Integration: problems

Integrals with a hidden symmetry

Paper 2 · Integration · Accessible · 20 marks

(i)
Let $a$ $>0$ , let $g$ be a continuous even function and let $h$ be a continuous odd function, both defined on $[-a,a]$ . By using the substitution $x$ $=-u$ , show that $\int_{-a}^{a}\frac{g(x)}{1+e^{h(x)}}\,\mathrm{d}x=\int_0^a g(x)\,\mathrm{d}x.$
7 marks
(ii)
Find $\displaystyle\int_{-1}^{1}\frac{1}{(1+x^2)(1+e^{x^3})}\,\mathrm{d}x$ .
3 marks
(iii)
Find $\displaystyle\int_{-\pi/3}^{\pi/3}\frac{\sec x}{1+e^{\tan x}}\,\mathrm{d}x$ .
4 marks
(iv)
By adapting the method of part (i) to the substitution $x$ $=\pi-u$ , find $\displaystyle\int_{0}^{\pi}\frac{\sin x}{1+e^{\cos x}}\,\mathrm{d}x$ .
6 marks

Paper 2 · Integration · Standard · 20 marks

For integers $n$ $\ge0$ , let $I_n=\int_0^{\pi/2}x\sin^nx\,\mathrm{d}x.$

(i)
Show that $\displaystyle\int_0^{\pi/2}\sin^{n-1}x\cos x\,\mathrm{d}x=\frac1n$ for $n$ $\ge1$ , and find $I_0$ and $I_1$ .
4 marks
(ii)
Show that, for $n$ $\ge2$ , $I_n=\frac{n-1}{n}I_{n-2}+\frac{1}{n^2}.$
8 marks
(iii)
Find $I_3$ and $I_4$ .
3 marks
(iv)
Using the substitution $x$ $=\frac{\pi}{2}-u$ , find $\displaystyle\int_0^{\pi/2}x\cos^4x\,\mathrm{d}x$ .
5 marks

Paper 2 · Integration · Standard · 20 marks

In this question you may assume that every improper integral that appears converges.

(i)
By using the substitution $x$ $=\frac1t$ , show that $\int_0^\infty\frac{1}{x^4-x^2+1}\,\mathrm{d}x=\int_0^\infty\frac{x^2}{x^4-x^2+1}\,\mathrm{d}x.$
4 marks
(ii)
By writing $\frac{x^2+1}{x^4-x^2+1}$ in terms of $u$ $=x-\frac1x$ , find $\displaystyle\int_0^\infty\frac{x^2+1}{x^4-x^2+1}\,\mathrm{d}x$ . Hence find the value of each integral in part (i).
7 marks
(iii)
Find constants $A$ , $B$ and $C$ such that $\frac{1}{x^6+1}=\frac{A}{x^2+1}+\frac{Bx^2+C}{x^4-x^2+1},$ and hence show that $\displaystyle\int_0^\infty\frac{1}{x^6+1}\,\mathrm{d}x=\frac{\pi}{3}$ .
6 marks
(iv)
Find $\displaystyle\int_0^\infty\frac{x^4}{x^6+1}\,\mathrm{d}x$ .
3 marks

Paper 2 · Integration · Standard · 20 marks

Let $\alpha$ be a constant with $0$ $<\alpha$ $<\pi$ .

(i)
Show that $\int_0^1\frac{1}{x^2+2x\cos\alpha+1}\,\mathrm{d}x=\frac{\alpha}{2\sin\alpha}.$
7 marks
(ii)
Deduce the values of $\displaystyle\int_0^1\frac{1}{x^2+x+1}\,\mathrm{d}x$ and $\displaystyle\int_0^1\frac{1}{x^2-x+1}\,\mathrm{d}x$ .
3 marks
(iii)
Show that $\frac{1}{x^4+x^2+1}=\frac12\left(\frac{x+1}{x^2+x+1}-\frac{x-1}{x^2-x+1}\right),$ and hence find $\displaystyle\int_0^1\frac{1}{x^4+x^2+1}\,\mathrm{d}x$ .
6 marks
(iv)
Find $\displaystyle\int_0^1\frac{x^2}{x^4+x^2+1}\,\mathrm{d}x$ .
4 marks

Paper 2 · Integration · Hard · 20 marks

(i)
Let $f$ be a continuous increasing function on $[0,1]$ and let $n$ be a positive integer. By considering the area under $y$ $=f(x)$ on each interval $\left[\frac{k-1}{n},\frac kn\right]$ , show that $\frac1n\sum_{k=0}^{n-1}f\!\left(\frac kn\right)\le\int_0^1 f(x)\,\mathrm{d}x\le\frac1n\sum_{k=1}^{n}f\!\left(\frac kn\right),$ and deduce that $0\le\frac1n\sum_{k=1}^{n}f\!\left(\frac kn\right)-\int_0^1f(x)\,\mathrm{d}x\le\frac{f(1)-f(0)}{n}.$
5 marks
(ii)
Show that $\displaystyle\int_0^1\ln(1+x^2)\,\mathrm{d}x=\ln2-2+\frac{\pi}{2}$ .
5 marks
(iii)
Let $P_n=\displaystyle\prod_{k=1}^{n}\left(1+\frac{k^2}{n^2}\right)$ . Show that $2e^{\frac{\pi}{2}-2}\le P_n^{1/n}\le 2^{1+\frac1n}e^{\frac{\pi}{2}-2}.$
6 marks
(iv)
Find $\displaystyle\lim_{n\to\infty}\frac{1}{n^2}\left(\prod_{k=1}^{n}(n^2+k^2)\right)^{1/n}$ .
4 marks

Paper 3 · Integration · Accessible · 20 marks

The curve $C$ is given parametrically by $x=t^2,\qquad y=t-\tfrac13t^3,\qquad t\in\mathbb{R}.$

(i)
Show that $C$ is symmetric about the $x$ -axis and that the points with $t$ $=\sqrt3$ and $t$ $=-\sqrt3$ coincide. Explain why the part of $C$ with $-\sqrt3$ $\le t$ $\le\sqrt3$ is a closed loop that does not cross itself.
4 marks
(ii)
Show that $\left(\frac{\mathrm{d}x}{\mathrm{d}t}\right)^2+\left(\frac{\mathrm{d}y}{\mathrm{d}t}\right)^2=(1+t^2)^2$ , and hence show that the length of the loop is $4\sqrt3$ .
5 marks
(iii)
Find the area of the region enclosed by the loop.
5 marks
(iv)
The region enclosed by the loop is rotated through $2\pi$ radians about the $x$ -axis. Find the volume of the solid formed.
6 marks

Paper 3 · Integration · Standard · 20 marks

(i)
Let $0$ $\le a$ $<b$ and let $f$ be a continuous function with $f(x)$ $\ge0$ on $[a,b]$ . The region $\{(x,y):a$ $\le x$ $\le b,\ 0$ $\le y$ $\le f(x)\}$ is rotated through $2\pi$ radians about the $y$ -axis. By dividing the region into thin strips parallel to the $y$ -axis, show that the volume of the solid formed is $2\pi\int_a^b xf(x)\,\mathrm{d}x.$
5 marks
(ii)
Let $c$ and $r$ be constants with $c$ $>r$ $>0$ . Using the substitution $x$ $=c+r\sin\theta$ , show that $\int_{c-r}^{c+r}x\sqrt{r^2-(x-c)^2}\,\mathrm{d}x=\frac{\pi cr^2}{2}.$
7 marks
(iii)
The disc $(x-c)^2+y^2$ $\le r^2$ , where $c$ $>r$ $>0$ , is rotated through $2\pi$ radians about the $y$ -axis to form a ring. Show that the volume of the ring is $2\pi^2cr^2$ .
4 marks
(iv)
Find the volume of the solid formed when the region $\frac{(x-4)^2}{4}$ ${}+\frac{y^2}{9}$ $\le1$ is rotated through $2\pi$ radians about the $y$ -axis.
4 marks

Paper 3 · Integration · Standard · 20 marks

The mean value of a continuous function $f$ on $[a,b]$ , where $a$ $<b$ , is $\displaystyle\bar f=\frac{1}{b-a}\int_a^b f(x)\,\mathrm{d}x$ .

(i)
Show that $\frac{1}{b-a}\int_a^b\big(f(x)-\bar f\big)^2\,\mathrm{d}x=\overline{f^2}-\bar f^{\,2},$ where $\overline{f^2}$ is the mean value of $\big(f(x)\big)^2$ on $[a,b]$ . Deduce that $\bar f^{\,2}$ $\le\overline{f^2}$ , with strict inequality if $f$ is not constant on $[a,b]$ .
5 marks
(ii)
Using the substitution $x$ $=\tan\theta$ , show that, for $0$ $<\beta$ $<\frac{\pi}{2}$ , $\int_0^{\tan\beta}\frac{1}{(1+x^2)^2}\,\mathrm{d}x=\frac{\beta}{2}+\frac{\sin2\beta}{4}.$
5 marks
(iii)
By applying (i) to $f(x)$ $=\frac{1}{1+x^2}$ on $[0,1]$ , show that $\pi<1+\sqrt5$ .
4 marks
(iv)
By applying (i) to the same function on the interval $\left[0,\frac{1}{\sqrt3}\right]$ , show that $\pi<\frac{\sqrt3+\sqrt{21}}{2}.$
6 marks

Paper 3 · Integration · Hard · 20 marks

The function $f$ is defined for $x$ $>0$ by $f(x)=\ln\left(\frac{e^x+1}{e^x-1}\right).$ The curve $C$ is $y$ $=f(x)$ , $x$ $>0$ .

(i)
Show that $f(x)$ $>0$ for $x$ $>0$ and that $f\big(f(x)\big)$ $=x$ for all $x$ $>0$ . State what this tells you about the curve $C$ .
4 marks
(ii)
Show that $\frac{\mathrm{d}y}{\mathrm{d}x}$ $=-\frac{1}{\sinh x}$ , and hence that $1+\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2=\coth^2x$ .
5 marks
(iii)
Find the coordinates of the point where $C$ meets the line $y$ $=x$ . Find also the length of the arc of $C$ between the points with $x$ $=\ln2$ and $x$ $=\ln3$ .
5 marks
(iv)
Show that $\sinh a\,\sinh f(a)$ $=1$ for every $a$ $>0$ . Deduce that, for $0$ $<a$ $<\ln(1+\sqrt2)$ , the arc of $C$ joining the points $\big(a,f(a)\big)$ and $\big(f(a),a\big)$ has length $-2\ln\sinh a$ , and check this against your answer to (iii).
6 marks