## Partial Fractions

### Partial Fractions

In solving algebra related problems and questions, we may sometimes deal with rational functions. A rational function is basically an algebraic polynomial fraction, in which we have polynomials on both the numerator and denominator.

Partial fractions is one of the simplest and most effective method in solving algebra related problems regarding rational functions.

In partial fractions, we separate the polynomials in our rational function into simpler form of polynomials.

Some of the applications of partial fractions include the solving of integration problems with rational functions, the binomial expansion and also the arithmetic series and sequences.

There are a few basic forms we need to memorize in partial fractions:

1.$$\enspace$$ The linear form:
$$\hspace{6em} {\small (ax + b)}$$
Example:

$${\large\frac{3x \ + \ 5}{(x \ + \ 1)(2x \ + \ 7)} \ \equiv \ \frac{A}{(x \ + \ 1)} + \frac{B}{(2x \ + \ 7)} }$$

2.$$\enspace$$ The quadratic form of a linear factor:
$$\hspace{6em} {\small (cx \ + \ d)^{2}}$$
Example:

$$\frac{3x + 5}{(x + 1){(2x + 7)}^{2}} \equiv \frac{A}{(x + 1)} + {\small\boxed{\frac{B}{(2x + 7)} + \frac{C}{{(2x + 7)}^{2}}}}$$

3.$$\enspace$$ The quadratic form that cannot be factorized:
$$\hspace{6em} {\small (c{x}^{2} \ + \ d) }$$
Example:

$${\large\frac{3x \ + \ 5}{(x \ + \ 1)(2{x}^{2} \ + \ 7)} \ \equiv \ \frac{A}{(x \ + \ 1)} + {\small\boxed{\frac{Bx \ + \ C}{(2{x}^{2} \ + \ 7)}}}}$$

After the polynomials in the denominator of the rational function is separated, make the denominators of the simpler terms to be the same. This is typically done by multiplying the denominators together .

To find each of the coefficients in the numerator (A, B or C), we can use substitution method or equating the coefficient method.

In the substitution method, we substitute a value of x that we freely choose in the left hand side numerator and the right hand side numerator and then find the coefficients one by one.

While in the equating the coefficient method, we expand the right hand side numerator and then compare each of the coefficients in the right hand side numerator with the left hand side numerator.

Both methods will be shown in the solution of the examples below. Give it a try and if you need any help, just look at the solution I have written. Cheers ! =) .
$$\\[1pt]$$

EXAMPLE:

$${\small 1.\enspace}$$ Let $${\small f(x) \ = \ {\large \frac{7{x}^{2} \ – \ 15x \ + \ 8}{(1 \ – \ 2x){(2 \ – \ x)}^{2} }} }$$
$$\\[1pt]$$
Express $${\small f(x) }$$ in partial fractions.
$$\\[1pt]$$

$$\\[1pt]$$
$${\small 2.\enspace}$$ Let $${\small f(x) \ = \ {\large \frac{ x \ – \ 4{x}^{2} }{(3 \ – \ x)(2 \ + \ {x}^{2}) }} }$$
$$\\[1pt]$$
Express $${\small f(x) }$$ in partial fractions.
$$\\[1pt]$$

$$\\[1pt]$$
$${\small 3.\enspace}$$ Let $${\small f(x) \ = \ {\large \frac{ 5{x}^{2} \ + \ x \ + \ 27 }{(2x \ + \ 1)( {x}^{2} \ + \ 9 ) }} }$$
$$\\[1pt]$$
Express $${\small f(x) }$$ in partial fractions.
$$\\[1pt]$$

$$\\[1pt]$$
$${\small 4.\enspace}$$ Let $${\small f(x) \ = \ {\large \frac{ 10 x \ + \ 9 }{(2x \ + \ 1){( 2x \ + \ 3 )}^{2} }} }$$
$$\\[1pt]$$
Express $${\small f(x) }$$ in partial fractions.
$$\\[1pt]$$

$$\\[1pt]$$
$${\small 5.\enspace}$$ Let $${\small f(x) \ = \ {\large \frac{ 2x(5 \ – \ x) }{(3 \ + \ x){( 1 \ – \ x )}^{2} }} }$$
$$\\[1pt]$$
Express $${\small f(x) }$$ in partial fractions.
$$\\[1pt]$$

$$\\[1pt]$$
$${\small 6.\enspace}$$ 9709/33/M/J/20 – Paper 33 June 2020 Pure Maths 3 No 7(a) and (b)
$$\\[1pt]$$
Let $${\small f(x) \ = \ {\large \frac{ 2 }{(2x \ – \ 1)( 2x \ + \ 1 ) }} }$$
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}$$ Express $${\small f(x) }$$ in partial fractions.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}$$ Using your answer to part (a), show that
$$\\[1pt]$$
$${\scriptsize {\Big( f(x) \Big)}^{2} \ = \ {\large \frac{ 1 }{ {(2x \ – \ 1)}^{2} }} \ – \ {\large \frac{ 1 }{ (2x \ – \ 1) }} }$$
$$\\[1pt]$$
$${\hspace{3em} \scriptsize \ + \ {\large \frac{ 1 }{ (2x \ + \ 1)}} \ + \ {\large \frac{ 1 }{ {(2x \ + \ 1)}^{2} }} . }$$
$$\\[1pt]$$

$$\\[1pt]$$
$${\small 7.\enspace}$$ 9709/32/F/M/21 – Paper 32 March 2021 Pure Maths 3 No 6(a)
$$\\[1pt]$$
Let $${\small f(x) \ = \ {\large \frac{ 5a }{(2x \ – \ a)( 3a \ – \ x ) }} }$$
$$\\[1pt]$$
Express $${\small f(x) }$$ in partial fractions.
$$\\[1pt]$$

$$\\[1pt]$$
$${\small 8.\enspace}$$ 9709/33/M/J/21 – Paper 33 June 2021 Pure Maths 3 No 4(a)
$$\\[1pt]$$
Let $${\small f(x) \ = \ {\large \frac{ 15 \ – \ 6x }{(1 \ + \ 2x)( 4 \ – \ x ) }} }$$
$$\\[1pt]$$
Express $${\small f(x) }$$ in partial fractions.
$$\\[1pt]$$

$$\\[1pt]$$
$${\small 9.\enspace}$$ 9709/32/M/J/21 – Paper 32 June 2021 Pure Maths 3 No 9(a)
$$\\[1pt]$$
Let $${\small f(x) \ = \ {\large \frac{ 14 \ – \ 3x \ + \ 2x^{2} }{(2 \ + \ x)( 3 \ + \ x^{2} ) }} }$$.
$$\\[1pt]$$
Express $${\small f(x) }$$ in partial fractions.
$$\\[1pt]$$

$$\\[1pt]$$
$${\small 10.\enspace}$$ 9709/31/M/J/21 – Paper 31 June 2021 Pure Maths 3 No 10
$$\\[1pt]$$
The variables $$x$$ and $$t$$ satisfy the differential equation:
$$\\[1pt]$$
$$\hspace{1.2em}{\large \frac{\mathrm{d}x}{\mathrm{d}t} } = x^{2} (1 + 2x)$$,
$$\\[1pt]$$
and $$x = 1$$ when $$t = 0$$.
$$\\[1pt]$$
Using partial fractions, solve the differential equation, obtaining an expression for $$t$$ in terms of $$x$$.
$$\\[1pt]$$

$$\\[1pt]$$

PRACTICE MORE WITH THESE QUESTIONS BELOW!

$${\small 1.\enspace}$$ Express $${\small {\large \frac{7{x}^{2} \ – \ 3x \ + \ 2}{x({x}^{2} \ + \ 1) }} }$$ in partial fractions.

$${\small 2. \enspace}$$ Let $${\small f(x) \ = \ {\large \frac{ 5{x}^{2} \ + \ x \ + \ 6 }{(3 \ – \ 2x)({x}^{2} \ + \ 4 )}} }$$
Express $${\small f(x) }$$ in partial fractions.

$${\small 3. \enspace}$$ Express $${\small {\large \frac{2 \ – \ x \ + \ 8{x}^{2}}{(1 \ – \ x)(1 \ + \ 2x)(2 \ + \ x) }} }$$ in partial fractions.

$${\small 4. \enspace}$$ Let $${\small f(x) \ = \ {\large \frac{ {x}^{2} \ + \ 3x \ + \ 3 }{(x \ + \ 1)(x \ + \ 3 )}} }$$
$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}$$ Express f(x) in partial fractions.
$${\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}$$ Hence show that,
$$\hspace{3em} {\small \displaystyle \int_{0}^{3} f(x) \ \mathrm{d}x = 3 \ – \ \frac{1}{2} \ln 2.}$$

$${\small 5. \enspace}$$ Let $${\small f(x) \ = \ {\large \frac{ {x}^{2} \ – \ 8x \ + \ 9 }{(1 \ – \ x){(2 \ – \ x )}^{2}}} }$$
Express $${\small f(x) }$$ in partial fractions.

$${\small 6. \enspace}$$ Let $${\small f(x) \ = \ {\large \frac{ 2{x}^{2} \ – \ 7x \ – \ 1 }{(x \ – \ 2)({x}^{2} \ + \ 3 )}} }$$
Express $${\small f(x) }$$ in partial fractions.

$$\\[12pt]{\small 7. \enspace}$$ Express $${\small {\large \frac{ x \ + \ 5 }{(x \ + \ 1)({x}^{2} \ + \ 3 )}} }$$ in the form:
$$\hspace{2em} {\large \frac{A}{( x \ + \ 1)} + \frac{Bx \ + \ C}{ ( {x}^{2} \ + \ 3) } }$$

$${\small 8. \enspace}$$ Express in partial fractions $${\small {\large \frac{{x}^{4}}{ {x}^{4} \ – \ 1 }} }$$

$${\small 9. \enspace}$$ Express in partial fractions $${\small {\large \frac{{x}^{3} \ + \ x \ – \ 1}{ {x}^{2} \ + \ {x}^{4} }} }$$

$$\\[10pt]{\small 10.\enspace}$$ Express in partial fractions
$$\hspace{2em}{\small {\large \frac{x \ + \ 5}{ {x}^{3} \ + \ 5{x}^{2} \ + \ 7x \ + \ 3 }} }$$

As always, if you have any particular questions to discuss, leave it in the comment section below. Cheers =) .

## Parametric Equations

### Parametric Equations

The traditional representation of y as a function of x or $${\small y = f(x)}$$ is inadequate to represent a curve or a surface.

To be able to draw a curve or a surface, we need to separate the x and y and write them in terms of an independent variable.

Parametric equations are used to express the Cartesian coordinates (x and y) in terms of another independent variable, usually named as t.

The typical procedure in this topic is to find the gradient of the curve or $${ \large\frac{\mathrm{d}y}{\mathrm{d}x} }$$. We can do this by finding each derivative of x and y with respect to t and then divide them both.

Let’s dig into some of the examples to show you what I mean. Cheers ! =) .
$$\\[1pt]$$

EXAMPLE:

$${\small 1.\enspace}$$ The parametric equations of a curve are
$$\\[1pt]$$
$$\hspace{3em} x = {\large \frac{1}{{\cos}^{3}t}}, \quad y = {\tan}^{3}t$$.
$$\\[1pt]$$
where $$0 \ \le \ t \ \le \ \frac{\pi}{2}$$
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(a\right). \enspace }$$ Show that $${\large\frac{\mathrm{d}y}{\mathrm{d}x} } = \sin \ t$$
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(b\right). \enspace }$$ Hence, show that the equation of the tangent to the curve at the point with parameter t is $$y = x \sin t \ – \ \tan t$$.

$$\\[1pt]$$
$${\small 2.\enspace}$$ The parametric equations of a curve are
$$\\[1pt]$$
$$\hspace{3em} x = e^{-t} \cos t, \quad y = e^{-t} \sin t$$.
$$\\[1pt]$$
Show that $${\large\frac{\mathrm{d}y}{\mathrm{d}x} } = \tan \big(t \ – \ {\large\frac{\pi}{4}}\big)$$

$$\\[1pt]$$
$${\small 3.\enspace}$$ The parametric equations of a curve are
$$\\[1pt]$$
$$\hspace{3em} x = \ln (2t \ + \ 3), \quad y = { \large\frac{3t \ + \ 2}{2t \ + \ 3} }$$.
$$\\[1pt]$$
Find the gradient of the curve at the point where it crosses the y-axis.

$$\\[1pt]$$
$${\small 4.\enspace}$$ The parametric equations of a curve are
$$\\[1pt]$$
$$\hspace{1.2em} {\small x = \ 2\sin \theta \ + \ \sin 2\theta, \enspace y = \ 2\cos \theta \ + \ \cos 2\theta }$$,
$$\\[1pt]$$
where $$0 \ \lt \ \theta \ \lt \ \pi$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(\textrm{i}\right). \enspace }$$ Obtain an expression for $${\small {\large\frac{\mathrm{d}y}{\mathrm{d}x} } }$$ in terms of $${\small \theta }$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(\textrm{ii}\right). \enspace }$$ Hence find the exact coordinates of the point on the curve at which the tangent is parallel to the y-axis.

$$\\[1pt]$$
$${\small 5.\enspace}$$ The parametric equations of a curve are
$$\\[1pt]$$
$$\hspace{1.2em} x = \ 2 t \ + \sin 2t, \enspace y = \ 1 \ – \ 2\cos 2t$$,
$$\\[1pt]$$
where $$-\frac{1}{2}\pi \ \lt \ t \ \lt \ \frac{1}{2}\pi$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(\textrm{i}\right). \enspace }$$ Show that $${\small {\large\frac{\mathrm{d}y}{\mathrm{d}x} } \ = \ 2 \tan t }$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(\textrm{ii}\right). \enspace }$$ Hence find the x-coordinate of the point on the curve at which the gradient of the normal is 2. Give your answer correct to 3 significant figures.

$$\\[1pt]$$
$${\small 6.\enspace}$$ 9709/33/M/J/20 – Paper 33 June 2020 Pure Maths 3 No 10(a), (b)
$$\\[1pt]$$

$$\\[1pt]$$
A tank containing water is in the form of a hemisphere. The axis is vertical, the lowest point is $$A$$ and the radius is $$r$$, as shown in the diagram. The depth of water at time $$t$$ is $$h$$.
$$\\[1pt]$$
At time $$t = 0$$ the tank is full and the depth of the water is $$r$$. At this instant a tap at $$A$$ is opened and water begins to flow out at a rate proportional to $$\sqrt{h}$$. The tank becomes empty at time $$t = 14$$.
$$\\[1pt]$$
The volume of water in the tank is $$V$$ when the depth is $$h$$. It is given that $$V = {\large\frac{1}{3}\pi} \big( 3r{h}^{2} \ – \ {h}^{3} \big)$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}$$ Show that $$h$$ and $$t$$ satisfy a differential equation of the form
$$\\[1pt]$$
$$\hspace{1.2em} {\large \frac{\mathrm{d}h}{\mathrm{d}t} } \ = \ – {\large\frac{ B }{ 2r{h}^{\frac{1}{2}} \ – \ {h}^{\frac{3}{2}} } }$$,
$$\\[1pt]$$
$$\hspace{1.2em}$$ where $$B$$ is a positive constant.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}$$ Solve the differential equation and obtain an expression for $$t$$ in terms of $$h$$ and $$r$$.

$$\\[1pt]$$
$${\small 7.\enspace}$$ 9709/33/M/J/21 – Paper 33 June 2021 Pure Maths 3 No 3(a), (b)
$$\\[1pt]$$
The parametric equations of a curve are
$$\\[1pt]$$
$$\hspace{1.2em} x = \ t \ + \ln (t+2), \enspace y = \ (t \ – \ 1) { \mathrm{e} }^{-2t}$$,
$$\\[1pt]$$
where $$t \ \gt \ -2$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}$$ Express $${\large\frac{\mathrm{d}y}{\mathrm{d}x} }$$in terms of $$t$$, simplifying your answer.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}$$ Find the exact $$y$$-coordinate of the stationary point of the curve.

$$\\[1pt]$$
$${\small 8.\enspace}$$ 9709/31/M/J/21 – Paper 31 June 2021 Pure Maths 3 No 6(a), (b)
$$\\[1pt]$$
The parametric equations of a curve are
$$\\[1pt]$$
$$\hspace{1.2em} x = \ln (2 \ + \ 3t), \enspace y = \ {\large\frac{t}{2 \ + \ 3t} }$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}$$ Show that the gradient of the curve is always positive.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}$$ Find the equation of the tangent to the curve at the point where it intersects the $$y$$-axis.

$$\\[1pt]$$
$${\small 9.\enspace}$$ 9709/32/F/M/22 – Paper 32 March 2022 Pure Maths 3 No 4
$$\\[1pt]$$
The parametric equations of a curve are
$$\\[1pt]$$
$$\hspace{1.2em} x = 1 \ – \ \cos \theta, \enspace y = \cos \theta \ – \ {\large\frac{1}{4} } \cos 2\theta$$.
$$\\[1pt]$$
Show that $${\small {\large\frac{\mathrm{d}y}{\mathrm{d}x}} = -2 \ \sin^{2} \big( {\large\frac{1}{2}}\theta \big) }$$.

$$\\[1pt]$$
$${\small 10.\enspace}$$ 9709/12/M/J/20 – Paper 12 June 2020 Pure Maths 1 No 3
$$\\[1pt]$$
A weather balloon in the shape of a sphere is being inflated by a pump. The volume of the balloon is increasing at a constant rate of 600 $${\small {\mathrm{cm}}^{3} }$$ per second. The balloon was empty at the start of pumping.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}$$ Find the radius of the balloon after 30 seconds.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}$$ Find the rate of increase of the radius after 30 seconds.

$$\\[1pt]$$
$${\small 11.\enspace}$$ 9709/12/O/N/19 – Paper 12 November 2019 Pure Maths 1 No 5
$$\\[1pt]$$

$$\\[1pt]$$
The diagram shows a solid cone which has a slant height of 15 cm and a vertical height of $${\small h}$$ cm.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(\mathrm{i}\right).\hspace{0.8em}}$$ Show that the volume, $${\small V \ {\mathrm{cm}}^{3} }$$, of the cone is given by $${\small V \ = \ {\large\frac{1}{3}}\pi (225h \ – \ {h}^{3}) }$$.
$$\\[1pt]$$
[The volume of a cone of radius $${\small r}$$ and vertical height $${\small r}$$ is $${\small {\large\frac{1}{3}}\pi {r}^{2} h }$$ ].
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(\mathrm{ii}\right).\hspace{0.8em}}$$ Given that $${\small h}$$ can vary, find the value of $${\small h}$$ for which $${\small V}$$ has a stationary value. Determine, showing all necessary working, the nature of this stationary value.

$$\\[1pt]$$

PRACTICE MORE WITH THESE QUESTIONS BELOW!

$${\small 1.\enspace}$$ The parametric equations of a curve are

$$x = \ \sin t \ + \cos t, \enspace y = \ {\sin}^{3}t \ + \ {\cos}^{3}t$$,

where $$\frac{\pi}{4} \ \le \ t \ \le \ \frac{5\pi}{4}$$

$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}$$ Show that $${ \large\frac{\mathrm{d}y}{\mathrm{d}x}} = \ -3 \ \sin t \ \cos t$$.
$${\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}$$ Find the gradient of the curve at the origin.
$${\small\hspace{1.2em}\left(c\right).\hspace{0.8em}}$$ Find the values of t for which the gradient of the curve is 1, giving your answers correct to 2 significant figures.

$${\small 2. \enspace}$$ The parametric equations of a curve are

$$x = \ a(2\theta \ – \ \sin 2\theta), \enspace y = \ a(1 \ – \ \cos 2\theta)$$.

Show that $${ \large\frac{\mathrm{d}y}{\mathrm{d}x}} = \ \cot \theta$$.

$${\small 3.\enspace}$$ The parametric equations of a curve are

$$x = \ \ln \cos \theta, \enspace y = \ 3\theta \ – \ \tan \theta$$,

where $$0 \ \le \ \theta \ \le \ \frac{1}{2}\pi$$.

$${\small\hspace{1.2em}\left(\textrm{i}\right).\hspace{0.8em}}$$ Express $${\small { \large\frac{\mathrm{d}y}{\mathrm{d}x}} }$$ in terms of $${\small \tan \theta}$$.
$${\small\hspace{1.2em}\left(\textrm{ii}\right).\hspace{0.6em}}$$ Find the exact y-coordinate of the point on the curve at which the gradient of the normal is equal to 1.

$${\small 4.\enspace}$$ The parametric equations of a curve are

$$x = \ {t}^{2} \ + 1, \enspace y = \ 4t \ + \ \ln \ (2t \ – \ 1)$$.

$${\small\hspace{1.2em}\left(\textrm{i}\right).\hspace{0.8em}}$$ Express $${\small { \large\frac{\mathrm{d}y}{\mathrm{d}x}} }$$ in terms of $${\small t}$$.
$${\small\hspace{1.2em}\left(\textrm{ii}\right).\hspace{0.6em}}$$ Find the equation of the normal to the curve at the point where $${\small t \ = \ 1}$$. Give your answer in the form ax + by + cz = 0.

$${\small 5.\enspace}$$ The parametric equations of a curve are

$$x = \ t \ + \cos t, \enspace y = \ \ln \ (1 + \sin t)$$,

where $$-\frac{1}{2}\pi \ \lt \ t \ \lt \ \frac{1}{2}\pi$$.

$${\small\hspace{1.2em}\left(\textrm{i}\right).\hspace{0.8em}}$$ Show that $${\small { \large\frac{\mathrm{d}y}{\mathrm{d}x}} \ = \ \sec t}$$.
$${\small\hspace{1.2em}\left(\textrm{ii}\right).\hspace{0.6em}}$$ Hence find the x-coordinates of the points on the curve at which the gradient is equal to 3. Give your answer correct to 3 significant figures.

$${\small 6.\enspace}$$ A curve has parametric equations

$$x = \ {t}^{2} \ + \ 3t \ + \ 1, \enspace y = \ {t}^{4} \ + \ 1$$.

The point P on the curve has parameter p. It is given that the gradient of the curve at P is 4. Show that $${\small p \ = \ \sqrt[3]{(2p \ + \ 3)} }$$.

$${\small 7.\enspace}$$ A curve has parametric equations

$$x = \ 1 \ + \ 2 \ \sin \theta \$$ and $$\ y = \ 4 \ + \ \sqrt{3} \ \cos \theta$$.

$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}$$ Find the equations of the tangent and normal at the point P where $${\small \theta \ = \ {\large\frac{\pi}{6}} }$$. Hence, find the area A of the triangle bounded by the tangent and normal at P, and the y-axis.
$${\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}$$ Determine the rate of change of xy at $${\small \theta \ = \ {\large\frac{\pi}{6}} }$$ if x increases at a constant rate of 0.1 units/s.

$${\small 8.\enspace}$$ A curve is defined parametrically by $$x \ = \ \frac{2t}{t + 1}$$ and $$y \ = \ \frac{{t}^{2}}{t + 1}$$.

$${\small\hspace{1.2em}\left(\textrm{i}\right).\hspace{0.8em}}$$ Find the equation of the normal to the curve at the point P(1, 1/2).
$${\small\hspace{1.2em}\left(\textrm{ii}\right).\hspace{0.6em}}$$ The normal at P meets the curve again at Q. Find the exact coordinates of Q.

$${\small 9.\enspace}$$ A curve has parametric equations:

$$x = \ \sec (\frac{\theta}{2}), \enspace y = \ \ln \ \sec (\frac{\theta}{2})$$,

where $$-\pi \ \lt \ \theta \ \lt \ \pi$$.

$${\small\hspace{1.2em}\left(\textrm{i}\right).\hspace{0.8em}}$$ Find the equation of the normal to the curve at the point at $${\small \theta \ = \ {\large\frac{\pi}{3}} }$$.
$${\small\hspace{1.2em}\left(\textrm{ii}\right).\hspace{0.6em}}$$ Determine the rate of change of x if the gradient of the curve at $${\small \theta \ = \ {\large\frac{\pi}{2}} }$$ is decreasing at a rate of 0.4 units per second.

$${\small 10.\enspace}$$ The parametric equations of a curve are

$$x = \ t \ + \ \ln t, \enspace y = \ t \ + \ {\textrm{e}}^{t}$$ for $$t \gt 0$$.

$${\small\hspace{1.2em}\left(\textrm{i}\right).\hspace{0.8em}}$$ Sketch the curve, indicating clearly all intercepts and asymptotes.
$${\small\hspace{1.2em}\left(\textrm{ii}\right).\hspace{0.6em}}$$ Show that, for all the points on the curve, $${\small { \large\frac{\mathrm{d}y}{\mathrm{d}x}} \ = \ { \large\frac{t(1 \ + \ {\textrm{e}}^{t})}{t \ + \ 1}} }$$.
Hence, deduce that the curve does not have any turning points.
$${\small\hspace{1.2em}\left(\textrm{iii}\right).\hspace{0.4em}}$$ Find, in exact form, the equation of the normal of the curve at the point where $${\small t \ = \ 1}$$.

As always, if you have any particular questions to discuss, leave it in the comment section below. Cheers =) .

## Integration and Differentiation

### Integration and Differentiation

Integration is an essential part of basic calculus. Algebra plays a very important part to become proficient in this topic.

I have compiled some of the questions that I have encountered during my Math tutoring classes. Do take your time to try the questions and learn from the solutions I have provided below. Cheers ! =) .

More Integration Exercises can be found here.

EXAMPLE:

$${\small 1.\enspace}$$ 9709/32/F/M/17 – Paper 32 Feb March 2017 Pure Maths 3 No 10
$$\\[1pt]$$

$$\\[1pt]$$
The diagram shows the curve $$\ {\small y \ = \ {(\ln x)}^{2} }$$. The x-coordinate of the point P is equal to e, and the normal to the curve at P meets the x-axis at Q.
$$\\[1pt]$$
$${\small\hspace{1.2em}(\textrm{i}).\hspace{0.7em}}$$ Find the x-coordinate of Q.
$$\\[1pt]$$
$${\small\hspace{1.2em}(\textrm{ii}).\hspace{0.7em}}$$ Show that $${\small \displaystyle \int \ln x \ \mathrm{d}x \ = \ x \ln x \ – \ x \ + \ c }$$, where c is a constant.
$$\\[1pt]$$
$${\small\hspace{1.2em}(\textrm{iii}).\hspace{0.5em}}$$ Using integration by parts, or otherwise, find the exact value of the area of the shaded region between the curve, the x-axis and the normal PQ.

$$\\[1pt]$$
$${\small 2.\enspace}$$ 9709/32/F/M/19 – Paper 32 Feb March 2019 Pure Maths 3 No 10
$$\\[1pt]$$

$$\\[1pt]$$
The diagram shows the curve $$\ {\small y \ = \ {\sin}^{3} x \sqrt{(\cos x)} \ }$$ for $$\ {\small 0 \leq x \leq \large{ \frac{1}{2}} \pi }$$, and its maximum point M.
$$\\[1pt]$$
$${\small\hspace{1.2em}(\textrm{i}).\hspace{0.7em}}$$ Using the substitution $${\small \ u \ = \ \cos x }$$, find by integration the exact area of the shaded region bounded by the curve and the x-axis.
$$\\[1pt]$$
$${\small\hspace{1.2em}(\textrm{ii}).\hspace{0.7em}}$$ Showing all your working, find the x-coordinate of M, giving your answer correct to 3 decimal places.

$$\\[1pt]$$
$${\small 3.\enspace}$$ 9709/32/M/J/20 – Paper 32 May June 2020 Pure Maths 3 No 9
$$\\[1pt]$$

$$\\[1pt]$$
The diagram shows the curves $$\ {\small \ y \ = \ \cos x \ }$$ and $$\ {\small \ y \ = \ \large{ \frac{k}{1 \ + \ x} } }$$, where k is a constant, for $$\ {\small \ 0 \leq x \leq \large{ \frac{1}{2}} \pi }$$. The curves touch at the point where x = p.
$$\\[1pt]$$
$${\small\hspace{1.2em}(\textrm{a}).\hspace{0.8em}}$$ Show that p satisfies the equation $${\small \ \tan p \ = \ \large{ \frac{1}{1 \ + \ p} } }$$.

$$\\[1pt]$$
$${\small 4.\enspace} \displaystyle \int_{1}^{a} \ln 2x \ \mathrm{d}x = 1.$$ Find $${\small a}$$.

$$\\[1pt]$$
$${\small 5.\enspace}$$ Use the substitution $$u = \sin 4x$$ to find the exact value of $$\displaystyle \int_{0}^{{\Large\frac{\pi}{24}}} \cos^{3} 4x \ \mathrm{d}x.$$

$$\\[1pt]$$
$${\small 6. \hspace{0.8em}(i).\hspace{0.8em}}$$ Use the trapezium rule with 3 intervals to estimate the value of: $$\displaystyle \int_{{\Large\frac{\pi}{9}}}^{{\Large\frac{2\pi}{3}}} \csc x \ \mathrm{d}x$$ giving your answer correct to 2 decimal places.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(ii\right).\hspace{0.8em}}$$ Using a sketch of the graph of $$y = \csc x$$, explain whether the trapezium rule gives an overestimate or an underestimate of the true value of the integral in part (i).

$$\\[1pt]$$
$${\small 7.\enspace}$$ Solve these integrations.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}} \displaystyle \int_{0}^{\infty} \frac{1}{{x}^{2} \ + \ 4} \ \mathrm{d}x$$
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(b\right).\hspace{0.8em}} \displaystyle \int_{0}^{3} \frac{1}{\sqrt{9 \ – \ {x}^{2}}} \ \mathrm{d}x$$
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(c\right).\hspace{0.8em}} \displaystyle \int_{-\infty}^{\infty} \frac{1}{9{x}^{2} \ + \ 4} \ \mathrm{d}x$$
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(d\right).\hspace{0.8em}} \displaystyle \int_{0}^{1} \frac{1}{\sqrt{x(1 \ – \ x)}} \ \mathrm{d}x$$
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(e\right).\hspace{0.8em}} \displaystyle \int_{1}^{\infty} \frac{1}{{(1 \ + \ x^2)}^{{\large\frac{3}{2}}}} \ \mathrm{d}x$$
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(f\right).\hspace{0.8em}} \displaystyle \int_{1}^{\infty} \frac{1}{x \sqrt{{x}^{2} \ – \ 1}} \ \mathrm{d}x$$

$$\\[1pt]$$
$${\small 8.\enspace}$$ The diagram shows the curve $${\small y = {e}^{{\large – \frac{1}{2}x}} \ \sqrt{(1 \ + \ 2x)}}$$ and its maximum point M. The shaded region between the curve and the axes is denoted by R.
$$\\[1pt]$$

$$\\[1pt]$$
$${\small \hspace{1.2em}(i). \enspace }$$ Find the x-coordinate of M.
$$\\[1pt]$$
$${\small \hspace{1.2em}(ii). \enspace }$$ Find by integration the volume of the solid obtained when R is rotated completely about the x-axis. Give your answer in terms of $${\small \pi}$$ and e.

$$\\[1pt]$$
$${\small 9.\enspace}$$ 9709/33/M/J/20 – Paper 33 June 2020 Pure Maths 3 No 2
$$\\[1pt]$$
Find the exact value of $$\displaystyle \int_{0}^{1} (2 \ – \ x) \mathrm{e}^{-2x} \ \mathrm{d}x$$.

$$\\[1pt]$$
$${\small 10.\enspace}$$ 9709/33/M/J/20 – Paper 33 June 2020 Pure Maths 3 No 7(a), (b), (c)
$$\\[1pt]$$
Let $${\small f(x) \ = \ {\large \frac{ 2 }{(2x \ – \ 1)( 2x \ + \ 1 ) }} }$$
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}$$ Express $${\small f(x) }$$ in partial fractions.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}$$ Using your answer to part (a), show that
$$\\[1pt]$$
$${\scriptsize {\Big( f(x) \Big)}^{2} \ = \ {\large \frac{ 1 }{ {(2x \ – \ 1)}^{2} }} \ – \ {\large \frac{ 1 }{ (2x \ – \ 1) }} }$$
$$\\[1pt]$$
$${\hspace{3em} \scriptsize \ + \ {\large \frac{ 1 }{ (2x \ + \ 1)}} \ + \ {\large \frac{ 1 }{ {(2x \ + \ 1)}^{2} }} . }$$
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(c\right).\hspace{0.8em}}$$ Hence show that $$\displaystyle \int_{1}^{2} {\Big( f(x) \Big)}^{2} \ \mathrm{d}x = \frac{2}{5} + \frac{1}{2} \ln \frac{5}{9}$$.

$$\\[1pt]$$
$${\small 11.\enspace}$$ 9709/32/M/J/20 – Paper 32 June 2020 Pure Maths 3 No 3
$$\\[1pt]$$
Find the exact value of $$\displaystyle \int_{1}^{4} x^{\frac{3}{2}} \ln x \ \mathrm{d}x$$.

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$${\small 12.\enspace}$$ 9709/32/M/J/20 – Paper 32 June 2020 Pure Maths 3 No 4
$$\\[1pt]$$
A curve has equation $$y = \cos x \sin 2x$$.
$$\\[1pt]$$
Find the $$x$$-coordinate of the stationary point in the interval $$0 \lt x \lt \frac{1}{2}\pi$$, giving your answer correct to 3 significant figures.

$$\\[1pt]$$
$${\small 13.\enspace}$$ 9709/32/M/J/20 – Paper 32 June 2020 Pure Maths 3 No 6(a), (b)
$$\\[1pt]$$

$$\\[1pt]$$
The diagram shows the curve $$y = {\large\frac{x}{1+{3x}^{4}} }$$, for $$x \geq 0$$, and its maximum point $$M$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}$$ Find the $$x$$-coordinate of $$M$$, giving your answer correct to 3 decimal places.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}$$ Using the substitution $$u = \sqrt{3}{x}^{2}$$, find by integration the exact area of the shaded region bounded by the curve, the $$x$$-axis and the line $$x = 1$$.

$$\\[1pt]$$
$${\small 14.\enspace}$$ 9709/32/M/J/20 – Paper 32 June 2020 Pure Maths 3 No 9(a)
$$\\[1pt]$$

$$\\[1pt]$$
The diagram shows the curves $$y = \cos x$$ and $$y = \frac{k}{1 \ + \ x}$$, where $$k$$ is a constant for $$0 \leq x \leq \frac{1}{2\pi}$$. The curves touch at the point where $$x = p$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}$$ Show that $$p$$ satisfies the equation $$\tan p = \frac{1}{1+p}$$.

$$\\[1pt]$$
$${\small 15.\enspace}$$ 9709/31/M/J/20 – Paper 31 June 2020 Pure Maths 3 No 4(a), (b)
$$\\[1pt]$$
The curve with equation $$y = { \mathrm{e} }^{2x} (\sin x + 3 \cos x)$$ has a stationary point in the interval $$0 \leq x \leq \pi$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}$$ Find the $$x$$-coordinate of this point, giving your answer correct to 2 decimal places.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}$$ Determine whether the stationary point is a maximum or a minimum.

$$\\[1pt]$$
$${\small 16.\enspace}$$ 9709/31/M/J/20 – Paper 31 June 2020 Pure Maths 3 No 5(a), (b)
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}$$ Find the quotient and remainder when $$2x^3 − x^2 + 6x + 3$$ is divided by $$x^2 + 3$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}$$ Using your answer to part (a), find the exact value of $$\displaystyle \int_{1}^{3} \frac{2x^3 \ – \ x^2 \ + \ 6x \ + \ 3}{x^2 \ + \ 3} \mathrm{d}x$$.

$$\\[1pt]$$
$${\small 17.\enspace}$$ 9709/31/M/J/20 – Paper 31 June 2020 Pure Maths 3 No 7(a), (b)
$$\\[1pt]$$
Let $$f(x) = { \large \frac{\cos x}{1 \ + \ \sin x} }$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}$$ Show that $$f'(x) \lt 0$$ for all $$x$$ in the interval $$-\frac{1}{2} \pi \lt x \lt \frac{3}{2} \pi$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}$$ Find $$\displaystyle \int_{\frac{\pi}{6}}^{\frac{\pi}{2} } f(x) \ \mathrm{d}x$$. Give your answer in a simplified exact form.

$$\\[1pt]$$
$${\small 18.\enspace}$$ 9709/32/F/M/21 – Paper 32 March 2021 Pure Maths 3 No 6(a), (b)
$$\\[1pt]$$
Let $${\small f(x) \ = \ {\large \frac{ 5a }{(2x \ – \ a)( 3a \ – \ x ) }} }$$
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}$$ Express $${\small f(x) }$$ in partial fractions.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}$$ Hence show that $$\displaystyle \int_{a}^{ 2a } f(x) \ \mathrm{d}x = \ln 6$$.

$$\\[1pt]$$
$${\small 19.\enspace}$$ 9709/32/F/M/21 – Paper 32 March 2021 Pure Maths 3 No 9(c)
$$\\[1pt]$$
Let $${\small f(x) \ = \ {\large \frac{ { \mathrm{e} }^{2x} \ + \ 1 }{{ \mathrm{e} }^{2x} \ – \ 1 }} }$$, for $$x \gt 0$$.
$$\\[1pt]$$
Find $$f'(x)$$. Hence find the exact value of x for which $$f'(x) = -8$$.

$$\\[1pt]$$
$${\small 20.\enspace}$$ 9709/32/F/M/21 – Paper 32 March 2021 Pure Maths 3 No 10(a), (b)
$$\\[1pt]$$

$$\\[1pt]$$
The diagram shows the curve $$y = \sin 2x \ {\cos}^{2} x$$ for $$0 \leq x \leq \frac{1}{2} \pi$$, and its maximum point $$M$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}$$ Using the substitution $$u = \sin x$$, find the exact area of the region bounded by the curve and the $$x$$-axis.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}$$ Find the exact $$x$$-coordinate of $$M$$.

$$\\[1pt]$$
$${\small 21.\enspace}$$ 9709/33/M/J/21 – Paper 33 June 2021 Pure Maths 3 No 4(a),(b)
$$\\[1pt]$$
Let $${\small f(x) \ = \ {\large \frac{ 15 \ – \ 6x }{(1 \ + \ 2x)( 4 \ – \ x ) }} }$$
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}$$ Express $${\small f(x) }$$ in partial fractions.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}$$ Hence find $$\displaystyle \int_{1}^{ 2 } f(x) \ \mathrm{d}x$$, giving your answer in the form $$\ln ( \frac{a}{b} )$$, where $$a$$ and $$b$$ are integers.

$$\\[1pt]$$
$${\small 22.\enspace}$$ 9709/33/M/J/21 – Paper 33 June 2021 Pure Maths 3 No 8(a),(b)
$$\\[1pt]$$

$$\\[1pt]$$
The diagram shows the curve $$y = {\large \frac{ \ln x }{ x^4 } }$$ and its maximum point $$M$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}$$ Find the exact $$x$$-coordinate of $$M$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}$$ By using integration by parts, show that for all $$a \gt 1, \displaystyle \int_{1}^{ a } \frac{ \ln x }{ x^4 } \ \mathrm{d}x \lt \frac{1}{9}$$.

$$\\[1pt]$$
$${\small 23.\enspace}$$ 9709/32/M/J/21 – Paper 32 June 2021 Pure Maths 3 No 4
$$\\[1pt]$$
Using integration by parts, find the exact value of $$\displaystyle \int_{0}^{ 2 } {\tan}^{-1} \big( \frac{1}{2} x \big) \ \mathrm{d}x$$.

$$\\[1pt]$$
$${\small 24.\enspace}$$ 9709/32/M/J/21 – Paper 32 June 2021 Pure Maths 3 No 6(a), (b)
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}$$ Prove that $$\mathrm{cosec} 2\theta − \cot 2\theta \equiv tan \theta$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}$$ Hence show that $$\displaystyle \int_{\frac{1}{4}\pi}^{ \frac{1}{3}\pi } (\mathrm{cosec} 2\theta − \cot 2\theta) \ \mathrm{d}\theta = \frac{1}{2} \ln 2$$.

$$\\[1pt]$$
$${\small 25.\enspace}$$ 9709/32/M/J/21 – Paper 32 June 2021 Pure Maths 3 No 8
$$\\[1pt]$$
The equation of a curve is $$y = { \mathrm{e} }^{-5x} \ {\tan}^{2} x$$ for $$-\frac{1}{2}\pi \lt x \lt \frac{1}{2}\pi$$.
$$\\[1pt]$$
Find the $$x$$-coordinates of the stationary points of the curve. Give your answers correct to 3 decimal places where appropriate.

$$\\[1pt]$$
$${\small 26.\enspace}$$ 9709/31/M/J/21 – Paper 31 June 2021 Pure Maths 3 No 7(a)
$$\\[1pt]$$

$$\\[1pt]$$
The diagram shows the curve $$y = {\large \frac{ {\tan}^{-1} x }{ \sqrt{x} } }$$ and its maximum point $$M$$ where $$x = a$$.
$$\\[1pt]$$
Show that a satisfies the equation $$a = \tan \Big( {\large \frac{2a}{1 \ + \ a^2} }\Big)$$.

$$\\[1pt]$$
$${\small 27.\enspace}$$ 9709/31/M/J/21 – Paper 31 June 2021 Pure Maths 3 No 9(a), (b)
$$\\[1pt]$$
The equation of a curve is $$y = { x^{-\frac{2}{3}}} \ \ln x \$$ for $$x \gt 0$$. The curve has one stationary point.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}$$ Find the exact coordinates of the stationary point.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}$$ Show that $$\displaystyle \int_{1}^{ 8 } y \ \mathrm{d}x = 18 \ln 2 \ – \ 9$$.

$$\\[1pt]$$
$${\small 28.\enspace}$$ 9709/32/F/M/22 – Paper 32 March 2022 Pure Maths 3 No 8(a), (b)
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}$$ Find the quotient and remainder when $${\small 8x^3 + 4x^2 + 2x + 7}$$ is divided by $${\small 4x^2 + 1}$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}$$ Hence find the exact value of $$\displaystyle \int_{0}^{ \frac{1}{2} } \frac{8x^3 + 4x^2 + 2x + 7}{4x^2 + 1} \ \mathrm{d}x$$.

$$\\[1pt]$$
$${\small 29.\enspace}$$ 9709/32/F/M/22 – Paper 32 March 2022 Pure Maths 3 No 11
$$\\[1pt]$$

$$\\[1pt]$$
The diagram shows the curve $$\ y \ = \ \sin x \ \cos 2x \$$ for $${\small \ 0 \le x \le {\large\frac{1}{2}}\pi}$$, and its maximum point $$M$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}$$ Find the $$x$$-coordinate of $$M$$, giving your answer correct to 3 significant figures.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}$$ Using the substitution $$\ u = \cos x$$, find the area of the shaded region enclosed by the curve and the $$x$$-axis in the first quadrant, giving your answer in a simplified exact form.

$$\\[1pt]$$
$${\small 30.\enspace}$$ 9709/13/O/N/21 – Paper 13 November 2021 Pure Maths 1 No 8
$$\\[1pt]$$

$$\\[1pt]$$
The diagram shows the curves with equations $$\ y \ = \ {x}^{ { \large – \frac{1}{2} } } \$$ and $$\ y \ = \ \frac{5}{2} \ – \ {x}^{ { \large \frac{1}{2} } }$$. The curves intersect at the points $$\ A( {\large\frac{1}{4}},2) \$$ and $$\ B( 4 , {\large\frac{1}{2}})$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}$$ Find the area of the region between the two curves.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}$$ The normal to the curve $$\ y \ = \ {x}^{ { \large – \frac{1}{2} } } \$$ at the point $$(1, 1)$$ intersects the $$y$$-axis at the point $$(0, p)$$. Find the value of $$p$$.

$$\\[1pt]$$
$${\small 31.\enspace}$$ 9709/13/O/N/21 – Paper 13 November 2021 Pure Maths 1 No 10
$$\\[1pt]$$
A curve has equation $$\ y = \mathrm{f}(x) \$$ and it is given that
$$\\[1pt]$$
$$\hspace{2em} \mathrm{f}^{\prime}(x) = { \big( \frac{1}{2}x \ + \ k \big) }^{-2} \ – \ { ( 1 \ + \ k ) }^{-2}$$,
$$\\[1pt]$$
where $${\small \ k \ }$$ is a constant. The curve has a minimum point at $${\small \ x \ = \ 2 }$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}$$ Find $$\ \mathrm{f}^{\prime\prime}(x)$$ in terms of $$k$$ and $$x$$, and hence find the set of possible values of $$k$$.
$$\\[1pt]$$
It is now given that $${\small \ k \ = \ −3 \ }$$ and the minimum point is at $${\small \ (2, \ 3\frac{1}{2}) }$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}$$ Find $$\ \mathrm{f}(x)$$.

$$\\[1pt]$$
$${\small 32.\enspace}$$ 9709/12/M/J/21 – Paper 12 June 2021 Pure Maths 1 No 9
$$\\[1pt]$$

$$\\[1pt]$$
The diagram shows part of the curve with equation $$\ {\small y^2 \ = \ x \ − \ 2 } \$$ and the lines $$\ {\small x \ = \ 5 } \$$ and $$\ {\small y \ = \ 1 }$$. The shaded region enclosed by the curve and the lines is rotated through $$\ {\small 360^{\circ} \ }$$ about the $$x$$-axis.
$$\\[1pt]$$
Find the volume obtained.

$$\\[1pt]$$
$${\small 33.\enspace}$$ 9709/12/M/J/21 – Paper 12 June 2021 Pure Maths 1 No 11
$$\\[1pt]$$
The gradient of a curve is given by
$$\\[1pt]$$
$$\hspace{2em} {\small {\large \frac{\mathrm{d}y}{\mathrm{d}x}} \ = \ 6{(3x \ – \ 5)}^{3} \ – \ k{x}^{2}}$$,
$$\\[1pt]$$
where $${\small \ k \ }$$ is a constant. The curve has a stationary point at $$\ {\small (2, -3.5) }$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}$$ Find the value of $$\ k$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}$$ Find the equation of the curve.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(c\right).\hspace{0.8em}}$$ Find $${\small {\large \frac{ {\mathrm{d}}^{2} y }{ \mathrm{d}{x}^{2}} } }$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(d\right).\hspace{0.8em}}$$ Determine the nature of the stationary point at $$\ {\small (2, -3.5) }$$.

$$\\[1pt]$$
$${\small 34.\enspace}$$ 9709/12/M/J/20 – Paper 12 June 2020 Pure Maths 1 No 8
$$\\[1pt]$$

$$\\[1pt]$$
The diagram shows part of the curve with equation $$\ {\small y \ = \ {\large\frac{6}{x}} }$$. The points $$\ {\small (1, 6) \ }$$ and $$\ {\small (3, 2) \ }$$ lie on the curve. The shaded region is bounded by the curve and the lines $$\ {\small y \ = \ 2 } \$$ and $$\ {\small x \ = \ 1 }$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}$$ Find the volume generated when the shaded region is rotated through $$\ {\small 360^{\circ} \ }$$ about the $$y$$-axis.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}$$ The tangent to the curve at a point $$X$$ is parallel to the line $$\ {\small y \ + \ 2x \ = \ 0 }$$. Show that $$X$$ lies on the line $$\ {\small y \ = \ 2x }$$.

$$\\[1pt]$$
$${\small 35.\enspace}$$ 9709/12/M/J/20 – Paper 12 June 2020 Pure Maths 1 No 10
$$\\[1pt]$$
The equation of a curve is
$$\\[1pt]$$
$$\hspace{2em} y \ = 54x \ – \ {(2x \ – \ 7)}^{3}$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}$$ Find $$\ {\small {\large \frac{\mathrm{d}y}{\mathrm{d}x}} \ }$$ and $${\small \ {\large \frac{ {\mathrm{d}}^{2} y }{ \mathrm{d}{x}^{2}} } }$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}$$ Find the coordinates of each of the stationary points on the curve.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(c\right).\hspace{0.8em}}$$ Determine the nature of each of the stationary points.

$$\\[1pt]$$
$${\small 36.\enspace}$$ 9709/12/O/N/19 – Paper 12 June 2019 Pure Maths 1 No 10
$$\\[1pt]$$

$$\\[1pt]$$
The diagram shows part of the curve $$\ {\small y \ = \ 1 \ – \ {\large\frac{4}{ {(2x \ + \ 1)}^{2} }} }$$. The curve intersects the $$x$$-axis at $$A$$. The normal to the curve at A intersects the $$y$$-axis at $$B$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(i\right).\hspace{0.8em}}$$ Obtain expressions for $$\ {\small {\large \frac{\mathrm{d}y}{\mathrm{d}x}} \ }$$ and $$\displaystyle \int y \ \mathrm{d}x$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(ii\right).\hspace{0.7em}}$$ Find the coordinates of $$B$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(iii\right).\hspace{0.6em}}$$ Find, showing all necessary working, the area of the shaded region.

$$\\[1pt]$$

PRACTICE MORE WITH THESE QUESTIONS BELOW!

$${\small 1.\enspace}$$ Find $$\displaystyle \int \frac{1}{x^2\sqrt{x^2 \ – \ 4}} \ \mathrm{d}x$$ using the substitution $${\small x \ = \ 2 \sec \theta }$$.

$${\small 2. \enspace}$$ Find the exact value of $$\displaystyle \int_{1}^{e} x^4 \ \ln \ x \ \mathrm{d}x$$.

$${\small 3. \enspace}$$ Find the exact value of $$\displaystyle \int_{4}^{10} \frac{2x \ + \ 1}{(x \ – \ 3)^2} \ \mathrm{d}x$$, giving your answer in the form of $${\small a \ + \ b \ \ln \ c}$$, where a, b and c are integers.

$${\small 4. \enspace}$$ Find the exact value of $$\displaystyle \int_{1}^{4} \frac{\ln \ x}{\sqrt{x}} \ \mathrm{d}x$$.

$${\small 5. \enspace}$$ Find the exact value of

$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}} \displaystyle \int_{0}^{\infty} {e}^{1 \ – \ 2x} \ \mathrm{d}x$$

$${\small\hspace{1.2em}\left(b\right).\hspace{0.8em}} \displaystyle \int_{-1}^{0} \big( 2 \ + \ \frac{1}{x \ – \ 1} \big) \ \mathrm{d}x$$

$${\small\hspace{1.2em}\left(c\right).\hspace{0.8em}} \displaystyle \int_{{\large\frac{\pi}{6}}}^{{\large \frac{\pi}{4}}} \cot x \ \mathrm{d}x$$

$${\small\hspace{1.2em}\left(d\right).\hspace{0.8em}}$$ Using your result in (c), find also the exact value of $$\displaystyle \int_{{\large\frac{\pi}{6}}}^{{\large \frac{\pi}{4}}} \csc 2x \ \mathrm{d}x$$ by using the identity $$\cot x \ – \ \cot 2x \ \equiv \ \csc 2x$$.

$${\small 6. \enspace}$$ The diagram shows the part of the curve $${\small y \ = \ f(x)}$$, where $${\small f(x) \ = \ p \ – \ {e}^{x} }$$ and p is a constant. The curve crosses the y-axis at (0, 2).

$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}$$ Find the value of p.

$${\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}$$ Find the coordinates of the point where the curve crosses the x-axis.

$${\small\hspace{1.2em}\left(c\right).\hspace{0.8em}}$$ What is the area of the shaded region R?

$${\small 7. \enspace}$$ Integrate the following:

$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}} \displaystyle \int \frac{x^2}{1 \ + \ {x}^{3}} \ \mathrm{d}x$$

$${\small\hspace{1.2em}\left(b\right).\hspace{0.8em}} \displaystyle \int x^4 \ \sin (x^5 \ + \ 2) \ \mathrm{d}x$$

$${\small\hspace{1.2em}\left(c\right).\hspace{0.8em}} \displaystyle \int e^{x} \ \sin x \ \mathrm{d}x$$

$${\small 8. \enspace}$$ Let $$I \ = \ \displaystyle \int_{0}^{1} {\large \frac{\sqrt{x}}{2 \ – \ \sqrt{x}}} \ \mathrm{d}x$$.

$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}$$ Using the substitution $${\small u = \ 2 \ – \ \sqrt{x}}$$, show that $$I \ = \ \displaystyle \int_{1}^{2} {\large \frac{2 {(2 \ – \ u)}^{2}}{u}} \ \mathrm{d}u$$.

$${\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}$$ Hence show that $$I \ = \ 8 \ \ln 2 \ – \ 5$$.

$${\small 9. \enspace}$$ The constant a is such that

$${\small\hspace{3em}} \displaystyle \int_{0}^{a} x{e}^{{\large \frac{1}{2}x}} \mathrm{d}x \ = \ 6$$.

Show that a satisfies the equation

$${\small\hspace{3em}} a \ = \ 2 \ + \ {e}^{{\large -\frac{1}{2}a}}$$.

$${\small 10. \enspace}$$ Use the substitution $${\small u \ = \ 1 \ + \ 3 \ \tan x }$$ to find the exact value of

$${\small\hspace{3em}} \ \displaystyle \int_{0}^{{\large\frac{\pi}{4}}} {\large \frac{\sqrt{1 \ + \ 3 \ \tan x}}{{\cos}^{2}x}} \ \mathrm{d}x$$.

As always, if you have any particular questions to discuss, leave it in the comment section below. Cheers =) .