Math topic中文

By substitution

Integration by a given substitution.

Practice questions

1Differentiate $\mathrm{e}^{x^2+2x}$ with respect to $x$ and hence find the exact value of $\int_{0}^{1} (x+1)\mathrm{e}^{x^2+2x} \; \mathrm{d}x$ …
2Show that $\cos 3\theta = 4\cos^{3}\theta - 3\cos\theta$ . Hence, or otherwise, evaluate $\displaystyle \int_{0}^{\pi/6} \sin 2\theta\,\cos 3\theta\, d\theta$ …
3Show that for $x>1$ , $\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\dfrac{1}{\sqrt{x^2-1}}\right)=-\dfrac{x}{(x^2-1)^{3/2}}\,.$ …
4Obtain a formula for $\displaystyle\int_{\frac{1}{2}}^{n} \frac{(\tan^{-1} 2x)^2}{1+4x^2}\,dx$ in terms of $n$ , where $n>0$ …
5A curve C is defined by the parametric equations $x = \theta^2 - 1,\quad y = \tan\frac{\theta}{2},\quad 0 \le \theta < \pi$ …
6By using the substitution $u = \sqrt{x+1}$ , find $\displaystyle \int \frac{\sqrt{x+1}}{x-1}\,dx$ . The region $R$ is boun…
7Solve the inequality $1-2\sin x>0$ for $0\le x\le \frac{\pi}{2}$ . Hence, evaluate the exact value of $\displaystyle \int_{0}^{\frac{\pi}{2}} \big|1-2\sin x\big|\,dx$ …
8Obtain a formula for $\int_{\frac{1}{2}}^{n} \frac{(\tan^{-1} 2x)^2}{1 + 4x^2} \,\mathrm{d}x$ in terms of $n$ , where $n > 0$ …
9[Cambridge/9233/2004] Use the substitution $t = \sqrt{(x + 1)}$ to find the exact value of $\int_{0}^{3} x \sqrt{(x + 1)} \, dx$ …
10A curve $C$ is defined parametrically by $x = a \tan t, \quad y = a \sec^2 t \sin t, \quad -\frac{\pi}{4} < t < \frac{\pi}{4},$ …
11The curve $C$ has equation $y=\ln\left(2-\lvert x\rvert\right)$ . The region $R$ is bounded by $C$ , the lines $y=\ln 2$ ,…
12A curve $C$ has equation $y = 2ax + \frac{b}{x}$ where $a$ and $b$ are non-zero real constants and $x \ne 0$ . Using diff…
13Find $\int x \ln(x^2 - 4) \text{d}x$ . Use the substitution $x = a \cos u$ to find $\int_{0}^{a} \sqrt{a^2 - x^2} \text{ d}x$ …
14By using the substitution $x = \cot \theta$ , for $0 < \theta < \frac{\pi}{4}$ , find $\displaystyle \int \frac{1}{x^2\sqrt{1+x^2}}\,\mathrm{d}x$ …
15By writing $\sin^3 x$ as $\sin^2 x(\sin x)$ , find $\int \sin^3 x \cos^3 x \text{ d}x$ . Find $\int \frac{5 - 6x}{\sqrt{5 - 3x^2 + 2x}} \text{ d}x$ …
16Find $\int \cot^2 3x \; \mathrm{d}x$ . Find the exact value of $\int_{2}^{4} \frac{x^2-4x+3}{x^2-4x+8} \; \mathrm{d}x$ . U…
17Find $\int \frac{x+2}{x^2+2x-3} \,\mathrm{d}x$ . Find $\int 2x\sin x \,\mathrm{d}x$ . Using the substitution $x = 3\sin\theta$ …
18Find $\int \frac{x}{\sqrt{2x-1}}\,dx.$ Using the substitution $t = \tan x$ , find $\int \frac{1}{4\cos^{2}x + 9\sin^{2}x}\,dx.$ …
19Find the exact value of the constant $p$ such that $\displaystyle \int_{5}^{p+9} \frac{1}{\sqrt{9-x}}\,dx = \displaystyle \int_{0}^{\tfrac{1}{4}} \frac{1}{\sqrt{1-4x^{2}}}\,dx.$ …
20The region $R$ is bounded by the lines $y=0$ , $y=\dfrac{5}{2}\sqrt{3}$ and the curve $x^2 = 25 - y^2$ . $R$ is rotated ab…
21By means of the substitution $x = \operatorname{cosec}\theta$ , show that $\int_{\frac{2}{\sqrt{3}}}^{2} \frac{1}{x^3 \sqrt{x^2 - 1}} \mathrm{d}x = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \sin^2(\theta) \mathrm{d}\theta$ …
22Find $\int \frac{1-3x}{1+9x^2} \, \text{d}x$ . Use the substitution $x = 2\sin\theta$ , where $\theta$ is acute, to find $\int \frac{(x-1)^2}{\sqrt{4-x^2}} \, \text{d}x$ …
23The diagram below shows a region $R$ bounded by the curve $(y+5)^{2}=x-3$ and the line $y=x-10$ . Find the volume of soli…
24Let $f(x)=\dfrac{x}{\sqrt{x+1}}$ . Show that $\displaystyle \int_{0}^{1} f(x)\,dx=\dfrac{2}{3}\bigl(2-\sqrt{2}\bigr)$ . Th…
25A curve $C$ is defined by the parametric equations $x = 2\cos\theta + 1, \quad y = \sin\theta, \quad \text{where } -\pi \le \theta \le \pi.$ …
26Curve C is a circle with radius 2 and center at the origin with equation $x^2 + y^2 = 4$ . Line L has the equation $\sqrt{3}y = x + 2$ …
27Use the substitution $t = e^{x}$ to show that $\int e^{2x}\,\tan^{-1}\bigl(e^{x}\bigr)\,dx = \int t\,\tan^{-1}(t)\,dt.$ …
28Find $\int \sin 3x\cos x\, dx$ . Find $\displaystyle \int \frac{x}{x^2 + 4x + 13}\, dx$ . Use the substitution $x = 3\sin \theta$ …
29Use the substitution $x=1+\sin u$ to find $\int\sqrt{2x-x^{2}}\,dx$ . The finite region $R$ , which includes the point (0…
30By using the substitution $x = \frac{1}{u}$ , show that $\int_{2\sqrt{2}}^{4} \frac{1}{x\sqrt{x^2 - 4}} \, \mathrm{d}x = \frac{\pi}{24}$ …
31Find $\int \frac{1}{9 - 4x^2} \text{ d}x$ . Using the substitution $\sqrt{x} = \cos\theta$ , find $\int \sqrt{\frac{x}{1-x}} \text{ d}x$ …
32Use the substitution $u = 1 - x^3$ to find $\int \frac{x^5}{\sqrt{1 - x^3}} \, \mathrm{d}x$ . Hence find $\int \frac{x^8}{\sqrt{1 - x^3}} \, \mathrm{d}x$ …
33[Cambridge/1997] Use the substitution $y = x^3$ to find the value of $\int_{0}^{2} \frac{x^2}{1 + x^6} \, dx.$
34[Cambridge/2003 modified] Use the substitution $x = \tan \theta$ to find $\int \frac{1 - x^2}{(1 + x^2)^2} \, dx.$
35Differentiate $\sqrt{x^2 - 1}$ with respect to $x$ . Hence find $\displaystyle \int x\cos^{-1}\left(\dfrac{1}{x}\right)\,dx$ …
36Find the following integrals. $\int \frac{\cos 3x - \text{cosec}^2 3x}{\sin 3x + \cot 3x} \text{ d}x$ $\int \frac{1-x}{\sqrt{1-16x^2}} \text{ d}x$ …
37用换元法求不定积分 $\displaystyle\int \frac{2x}{x^2 + 1} \, dx$ 。
38Using integration by parts, or otherwise, find $\displaystyle \int \frac{1}{y}\ln y\,dy$ . Given that $y>1$ and $\displaystyle \frac{dy}{dx}=\frac{y}{\ln y}$ …
39Do not use a graphic calculator in answering this question. It is given that $\sin x > \dfrac{2x}{\pi}$ for $0 < x < \dfrac{\pi}{2}$ …
40Find $\int \tan^2(x - 1) \text{ d}x ,$ $\int \sin^{-1} 2x \text{ d}x .$
41Find the exact value of the following integrals. Find the exact value of $\int_{\frac{\sqrt{2}}{2}}^{\frac{\sqrt{3}}{2}} \frac{2\sin^{-1} x}{\sqrt{1-x^2}}\,dx .$ …
42A curve P is defined by $y = -\sin^2 x$ where $-\pi \le x \le 0$ . The finite region R is bounded by the curve P, the y-a…
43Using integration by parts, or otherwise, find $\displaystyle\int \frac{1}{y}\ln y\,dy$ . Given that $y>1$ and $\displaystyle\frac{dy}{dx}=\frac{y}{\ln y}$ …
44State the constants $A$ and $B$ such that, for all values of $x$ , $11 - 4x = A(4 - 2x) + B.$ Hence, find $\int \frac{11 - 4x}{\sqrt{x(4 - x)}} \, \mathrm{d}x$ …
45A curve $C$ is defined by the parametric equations $x = \theta^2 - 1$ , $y = \tan\frac{\theta}{2}$ , $0 \le \theta < \pi$ .…
46The curve $C$ has equation $y=\ln\left(2-\lvert x\rvert\right)$ . The region $R$ is bounded by $C$ , the lines $y=\ln 2$ ,…
47Curve $C$ is a circle with radius 2 and center at the origin with equation $x^2 + y^2 = 4$ . Line $L$ has the equation $\sqrt{3}y = x + 2$ …
48By using the substitution $u = 1 + t^3$ , find the exact value of $\int_{0}^{2} \frac{t^5}{\left(1 + t^3\right)^3} \text{ d}t$ …
49The curve $C$ is defined by the parametric equations $x = 1 - \cos t, \quad y = \sin 2t, \quad \text{where } 0 \le t \le \frac{\pi}{2}.$ …
50A curve $C$ has parametric equations $x = 2\cos t + 1, \quad y = \sin t , \quad 0 \le t < 2\pi .$ Sketch $C$ , indicati…
51Solve the inequality $1-2\sin x>0$ for $0 \le x \le \frac{\pi}{2}$ . Hence, evaluate the exact value of $\int_{0}^{\pi/2} |1-2\sin x|\,dx.$ …
52A function f is defined as $f(x)=\ln(5-x)$ , $x<5$ . Find the volume of revolution when the region bounded by the curve of…
53The diagram shows the curve $C$ with parametric equations $x = (1+t)^2, \quad y = (3-t)^2.$ The curve $C$ meets the ax…
54Using standard series from the List of Formula (MF26), expand $\ln(k + 2x)^n$ in ascending powers of $x$ , as far as the…
55Using the substitution $u = \cos x$ , find $\displaystyle \int \frac{\sin 2x}{\cos^2 x + \cos x + 2}\,dx$ , giving your an…
56Find $\int \sin 3x \cos x \text{ d}x$ . Find $\int \frac{x}{x^2 + 4x + 13} \text{ d}x$ . Use the substitution $x = 3\sin\theta$ …
57Obtain a formula for $\displaystyle\int_{\tfrac{1}{2}}^{n} \frac{(\tan^{-1}2x)^2}{1+4x^2}\,dx$ in terms of $n$ , where $n>0$ …
58By using the substitution $x = a\tan\theta$ , show that $\displaystyle \int \frac{1}{x\sqrt{x^2 + a^2}}\,dx = \frac{1}{a}\ln\frac{x}{\sqrt{x^2 + a^2} + a} + c ,$ …
59Find $\int \frac{\cos(\ln x)}{x^2} \, \mathrm{d}x$ . Using the substitution $u = \sqrt{x+3}$ , find $\int_1^6 \frac{x-2}{x\sqrt{x+3}} \, \mathrm{d}x$ …
60Find $\int \frac{3 - 2x}{\sqrt{5 + 4x - x^2}} \, \mathrm{d}x$ . Show that $\int_{\sqrt{2}}^{4} |x - 3| \, \mathrm{d}x = a - b\sqrt{2},$ …