## 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 ! =) .
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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.
$$\\[1pt]$$

$$\\[1pt]$$

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
$$\\[1pt]$$
$${\small x = \ 2\sin \theta \ + \ \sin 2\theta, \enspace y = \ 2\cos \theta \ + \ \cos 2\theta }$$,
$$\\[1pt]$$
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
$$\\[1pt]$$
$$x = \ 2 t \ + \sin 2t, \enspace y = \ 1 \ – \ 2\cos 2t$$,
$$\\[1pt]$$
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.

$$\\[1pt]$$

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 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 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 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.

$$\\[1pt]$$

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 =) .