# 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
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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.
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$${\small\hspace{1.2em}(\textrm{i}).\hspace{0.7em}}$$ Find the x-coordinate of Q.
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$${\small\hspace{1.2em}(\textrm{ii}).\hspace{0.7em}}$$ Show that $${\small \displaystyle \int \ln x \ \mathrm{d}x \ = \ x \ln x \ – \ x \ + \ c }$$, where c is a constant.
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$${\small\hspace{1.2em}(\textrm{iii}).\hspace{0.5em}}$$ 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.

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$${\small 2.\enspace}$$ 9709/32/F/M/19 – Paper 32 Feb March 2019 Pure Maths 3 No 10
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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.
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$${\small\hspace{1.2em}(\textrm{i}).\hspace{0.7em}}$$ 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.
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$${\small\hspace{1.2em}(\textrm{ii}).\hspace{0.7em}}$$ Showing all your working, find the x-coordinate of M, giving your answer correct to 3 decimal places.

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$${\small 3.\enspace}$$ 9709/32/M/J/20 – Paper 32 May June 2020 Pure Maths 3 No 9
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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.
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$${\small\hspace{1.2em}(\textrm{a}).\hspace{0.8em}}$$ Show that p satisfies the equation $${\small \ \tan p \ = \ \large{ \frac{1}{1 \ + \ p} } }$$.

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

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

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$${\small 6. \hspace{0.8em}(i).\hspace{0.8em}}$$ 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.
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$${\small\hspace{1.2em}\left(ii\right).\hspace{0.8em}}$$ 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).

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

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$${\small 8.\enspace}$$ 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.
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$${\small \hspace{1.2em}(i). \enspace }$$ Find the x-coordinate of M.
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$${\small \hspace{1.2em}(ii). \enspace }$$ 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.

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

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

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$${\small 11.\enspace}$$ 9709/32/M/J/20 – Paper 32 June 2020 Pure Maths 3 No 3
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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
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A curve has equation $$y = \cos x \sin 2x$$.
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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.

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$${\small 13.\enspace}$$ 9709/32/M/J/20 – Paper 32 June 2020 Pure Maths 3 No 6(a), (b)
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The diagram shows the curve $$y = {\large\frac{x}{1+{3x}^{4}} }$$, for $$x \geq 0$$, and its maximum point $$M$$.
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$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}$$ Find the $$x$$-coordinate of $$M$$, giving your answer correct to 3 decimal places.
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$${\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}$$ 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 $$x = 1$$.

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$${\small 14.\enspace}$$ 9709/32/M/J/20 – Paper 32 June 2020 Pure Maths 3 No 9(a)
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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 $$x = p$$.
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$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}$$ Show that $$p$$ satisfies the equation $$\tan p = \frac{1}{1+p}$$.

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

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

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

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

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

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$${\small 20.\enspace}$$ 9709/32/F/M/21 – Paper 32 March 2021 Pure Maths 3 No 10(a), (b)
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The diagram shows the curve $$y = \sin 2x \ {\cos}^{2} x$$ for $$0 \leq x \leq \frac{1}{2} \pi$$, and its maximum point $$M$$.
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$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}$$ Using the substitution $$u = \sin x$$, find the exact area of the region bounded by the curve and the $$x$$-axis.
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$${\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}$$ Find the exact $$x$$-coordinate of $$M$$.

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$${\small 21.\enspace}$$ 9709/33/M/J/21 – Paper 33 June 2021 Pure Maths 3 No 4(a),(b)
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Let $${\small f(x) \ = \ {\large \frac{ 15 \ – \ 6x }{(1 \ + \ 2x)( 4 \ – \ x ) }} }$$
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$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}$$ Express $${\small f(x) }$$ in partial fractions.
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$${\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}$$ 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.

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$${\small 22.\enspace}$$ 9709/33/M/J/21 – Paper 33 June 2021 Pure Maths 3 No 8(a),(b)
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The diagram shows the curve $$y = {\large \frac{ \ln x }{ x^4 } }$$ and its maximum point $$M$$.
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$${\small\hspace{1.2em}\left(a\right).\hspace{0.8em}}$$ Find the exact $$x$$-coordinate of $$M$$.
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$${\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}$$ By using integration by parts, show that for all $$a \gt 1, \displaystyle \int_{1}^{ a } \frac{ \ln x }{ x^4 } \ \mathrm{d}x \lt \frac{1}{9}$$.

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

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

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

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$${\small 26.\enspace}$$ 9709/31/M/J/21 – Paper 31 June 2021 Pure Maths 3 No 7(a)
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The diagram shows the curve $$y = {\large \frac{ {\tan}^{-1} x }{ \sqrt{x} } }$$ and its maximum point $$M$$ where $$x = a$$.
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Show that a satisfies the equation $$a = \tan \Big( {\large \frac{2a}{1 \ + \ a^2} }\Big)$$.

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

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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 =) .

##### 1 comment
1. Please can you solve question 4 of Oct/Nov 21-Variant-31