Math topic中文

Standard integrands

Integration of the following standard integrands:

$\dfrac{\mathrm{f}'(x)}{\mathrm{f}(x)}$ (including $n=-1$ giving $\ln$ )
$\mathrm{f}'(x)e^{\mathrm{f}(x)}$
$\sin^2 x$ , $\cos^2 x$ , $\tan^2 x$
$\dfrac{1}{a^2+x^2}$ , $\dfrac{1}{a^2-x^2}$ , $\dfrac{1}{\sqrt{a^2-x^2}}$ and $\dfrac{1}{x^2-a^2}$

Practice questions

1Differentiate $\mathrm{e}^{x^2+2x}$ with respect to $x$ and hence find the exact value of $\int_{0}^{1} (x+1)\mathrm{e}^{x^2+2x} \; \mathrm{d}x$ …
25 (a) Solve the following integral. $\displaystyle\int \frac{1-2x}{\sqrt{3+2x-x^2}}\,\mathrm{d}x.$ $\displaystyle\int x\ln\left(2-x^2\right)\,\mathrm{d}x$ …
3[Cambridge/9709/2010] The diagram shows parts of the curves $y = 9 - x^3$ and $y = \frac{8}{x^3}$ and their points o…
4Show that $\cos 3\theta = 4\cos^{3}\theta - 3\cos\theta$ . Hence, or otherwise, evaluate $\displaystyle \int_{0}^{\pi/6} \sin 2\theta\,\cos 3\theta\, d\theta$ …
5Find $\displaystyle \int \frac{3e^x}{5 - 0.3e^x} \, dx$ $\displaystyle \int \cos(\ln x) \, dx$ , where $x > 0$ the exact…
6Given that the first three terms in the expansion of $\dfrac{1}{a+bx}$ are $\dfrac{1}{2}+\dfrac{1}{4}x+cx^2$ , find the v…
7Show that for any complex number $z = e^{i\theta}$ , $z^{n} + \dfrac{1}{z^{n}} = 2\cos n\theta .$ By taking $n = 1$ , sh…
8A curve with equation $y=f(x)$ is transformed by a reflection in the y-axis, followed by a translation of 1 unit in the…
9Show that for $x>1$ , $\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\dfrac{1}{\sqrt{x^2-1}}\right)=-\dfrac{x}{(x^2-1)^{3/2}}\,.$ …
10The curve $C$ is defined by the parametric equations $x = t - \sin t$ , $y = \sin^{2} t$ , $\;0 \le t \le \pi$ . The point…
11Sketch the graphs of $y = 3e^x$ and $y = x + 3$ on the same diagram. Indicate clearly the coordinates of the points of i…
12It is given that $f(x)=\dfrac{1}{4+9x^2}$ . Find $\displaystyle\int f(x)\,dx$ . Find the binomial expansion for $f(x)$ , up…
13The points $(x,y)$ on a curve satisfy the equations $x = \tfrac{1}{2}t^2 - 2t$ and $\dfrac{dy}{dx} = \dfrac{1}{2t - 1}$ …
14Let f(x) be a cubic polynomial where f(2) = 6 and f(-1) = 15. Also, f(x) takes on stationary values at x = -2 and x = 1.…
15In a large town, the number of people infected by a particular virus $t$ days after the virus was first discovered is $x$ …
16Consider the second order differential equation $\frac{d^{2}x}{dt^{2}} + a\frac{dx}{dt} = b,$ where $a$ and $b$ are no…
17Write down constants $A$ and $B$ such that for all values of $x$ , $x - 1 = A(2x - 4) + B.$ By expressing $x^2 - 4x + 5$ …
18You are given that $C = \int_{0}^{\pi} \mathrm{e}^{-2x} \cos x \; \mathrm{d}x$ and $S = \int_{0}^{\pi} \mathrm{e}^{-2x} \sin x \; \mathrm{d}x$ …
19A 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…
20By using the substitution $u = \sqrt{x+1}$ , find $\displaystyle \int \frac{\sqrt{x+1}}{x-1}\,dx$ . The region $R$ is boun…
21The point $P$ travels along the curve $C$ with equation $y = x \sin^{-1} x$ , $-1 < x < 1$ . Let the gradient of the curve…
22Sketch the curve with equation $\frac{x^2}{400} + \frac{(y-10)^2}{900} = 1.$ [There is no need to indicate the x-inter…
23Solve the inequality $1-2\sin x>0$ for $0\le x\le \frac{\pi}{2}$ . Hence, evaluate the exact value of $\displaystyle \int_{0}^{\frac{\pi}{2}} \big|1-2\sin x\big|\,dx$ …
24[Cambridge/9233/2004] Use the substitution $t = \sqrt{(x + 1)}$ to find the exact value of $\int_{0}^{3} x \sqrt{(x + 1)} \, dx$ …
25Find $\displaystyle\int x^{2}e^{-2x}\,\mathrm{d}x$ . The curve with equation $y = xe^{-x}$ and the line with equation $y=\tfrac{1}{e}x$ …
26A farmer has a plot of the land, $OAB$ , with adjacent sides $OA$ and $OB$ . With reference to $O$ as the origin, $OA$ and…
27A curve $C$ has equation $y = 2ax + \frac{b}{x}$ where $a$ and $b$ are non-zero real constants and $x \ne 0$ . Using diff…
28In a large town, the number of people infected by a particular virus $t$ days after the virus was first discovered is $x$ …
29The variables $x$ and $y$ are related by the differential equation $\pi \dfrac{dy}{dx} + y(3 - \pi \tan x) = 0$ . Using t…
30In harvesting of renewable natural resources, it is desirable that policies are formulated to allow maximal harvest of t…
31Find $\displaystyle \int \sin 3x \cos x\, dx$ . Find $\displaystyle \int e^{2x} \cos 3x\, dx$ .
32The function $\text{f}$ is defined by $\text{f} : x \to 2x - \frac{1}{2x}, \quad 0 < x < 2.$ It is given that $\text{f}^{-1}$ …
33Given that $z = e^{2x}\frac{dy}{dx}$ , express $\frac{dz}{dx}$ in terms of $x$ , $\frac{dy}{dx}$ and $\frac{d^{2}y}{dx^{2}}$ …
34A family of curves is defined by the differential equation $xy\,\frac{dy}{dx} = x^{2} + y^{2}.$ By substituting $y = ux$ …
35Find $\int x \ln(x^2 - 4) \text{d}x$ . Use the substitution $x = a \cos u$ to find $\int_{0}^{a} \sqrt{a^2 - x^2} \text{ d}x$ …
36In military exercises, parachutists jump from stationary helicopters and their motion is tracked by sensors tagged to th…
37Find $\int x^{2} e^{-2x}\, dx$ . The curve with equation $y = x e^{-x}$ and the line with equation $y = \tfrac{1}{e} x$ …
38By using the substitution $x = \cot \theta$ , for $0 < \theta < \frac{\pi}{4}$ , find $\displaystyle \int \frac{1}{x^2\sqrt{1+x^2}}\,\mathrm{d}x$ …
39Show that $\dfrac{d}{dx}(\tan x)=\sec^2 x$ Show that $\sin 2x\cot x = 2\cos^2 x$ Hence, find the exact value of $\displaystyle\int_{\tfrac{\pi}{30}}^{\tfrac{\pi}{5}} \csc(10x)\tan(5x)\,dx$ …
40[Cambridge/N78/I/20] Find $\frac{\text{d}}{\text{d}x}(\tan^{-1}(x^2))$ and hence, or otherwise, find $\int x \tan^{-1}(x^2) \, dx$ …
41By writing $\sin^3 x$ as $\sin^2 x(\sin x)$ , find $\int \sin^3 x \cos^3 x \text{ d}x$ . Find $\int \frac{5 - 6x}{\sqrt{5 - 3x^2 + 2x}} \text{ d}x$ …
42Find $\int \cot^2 3x \; \mathrm{d}x$ . Find the exact value of $\int_{2}^{4} \frac{x^2-4x+3}{x^2-4x+8} \; \mathrm{d}x$ . U…
43Find $\int \sin px \cos qx \text{ d}x$ , where $p$ and $q$ are constants such that $p \ne q$ and $p \ne -q$ . Given that $m \ne 0$ …
44Find the exact value of $\pi \int_{1}^{6} x^3 \text{ d}x$ . The diagram shows a sketch of the curve $y = x^{\frac{3}{2}}$ …
45Find $\int \frac{x+2}{x^2+2x-3} \,\mathrm{d}x$ . Find $\int 2x\sin x \,\mathrm{d}x$ . Using the substitution $x = 3\sin\theta$ …
46Find $\int \frac{x}{\sqrt{2x-1}}\,dx.$ Using the substitution $t = \tan x$ , find $\int \frac{1}{4\cos^{2}x + 9\sin^{2}x}\,dx.$ …
47Find $\displaystyle \int \dfrac{6+2x}{\sqrt{1-4x-x^2}}\,\mathrm{d}x$ . The diagram above shows the region $R$ bounded by…
48The Stoke's Law states that when a spherical object of radius $R$ is falling at velocity $v$ in a fluid, the resistance…
49Find $\displaystyle \int x \sin^{2} x \, dx$ . The region R is bounded by the curve $y = \sqrt{x \sin x}$ , the lines x =…
50Find the exact value of the constant $p$ such that $\displaystyle \int_{5}^{p+9} \frac{1}{\sqrt{9-x}}\,dx = \displaystyle \int_{0}^{\tfrac{1}{4}} \frac{1}{\sqrt{1-4x^{2}}}\,dx.$ …
51The variables $x$ and $y$ are related by the differential equation $\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} = 2\mathrm{e}^{-2x} .$ …
52It is given that $f(x)=\ln(a+x)$ , $x\in\mathbb{R}$ , $x>-a$ , where $a$ is a constant. Using the standard series from the…
53By means of the substitution $x = \operatorname{cosec}\theta$ , show that $\int_{\frac{2}{\sqrt{3}}}^{2} \frac{1}{x^3 \sqrt{x^2 - 1}} \mathrm{d}x = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \sin^2(\theta) \mathrm{d}\theta$ …
54By using the substitution $u = x - y$ , solve the differential equation $\dfrac{dy}{dx} + 4(x - y)^2\cos^2 x = \sin^2 x.$ …
55Find the area of the region bounded by the graphs of $y = 2x^2 + 3$ and $y = -2x + 7$ . State the area of the region boun…
56Integrate by parts twice to show that $\int e^{2x}\sin x\,dx=\tfrac{1}{5}e^{2x}(2\sin x-\cos x)+c.$ The diagram below…
57A chemical substance Uru starts to form when $N$ units of Vibranium react with $M$ units of Kryptonite, where $M < N$ . L…
58On the same axes, sketch the graphs of $y = \sin 2x$ and $y = \cos x$ for $0 \le x \le \frac{\pi}{2}$ , stating the exact…
59It is given that $2xy\dfrac{dy}{dx} = x^{2} - y^{2}$ where $x>0$ , $y<0$ . Using the substitution $u = \dfrac{y}{x}$ , show…
60In a chemical reaction, a compound X is formed from two compounds Y and Z. The masses in grams of X, Y and Z present at…