Math topic中文

Integration reverse

integration as the reverse of differentiation

Practice questions

1Given that $y = \dfrac{3\ln 2x}{x^2}$ for $x>0$ . Show that $\dfrac{dy}{dx} = \dfrac{3}{x^3}\left(1 - 2\ln 2x\right).$ He…
2The equation of a curve is given by $y = 5xe^{2x+1}$ . Find $\frac{dy}{dx}$ . Hence, find $\int xe^{2x+1} \text{ d}x$ . Det…
3It is given that $f'(x)=6\cos 3x-12\sin 3x$ and $f\left(\frac{\pi}{3}\right)=0$ . Find $f(x)$ . Find the exact value of $f''\left(\frac{\pi}{12}\right)$ …
4The gradient function of a curve $y = f(x)$ is given by $\frac{36}{(2x+1)^2}$ for $x > -\frac{1}{2}$ . The curve passes t…
5It is given that $y = 4x \sin 2x$ . Show that $\frac{\mathrm{d}y}{\mathrm{d}x}$ can be expressed in the form $ax \cos 2x + b \sin 2x$ …
6Express $\dfrac{2x^{2}+3x}{x^{2}+3x+2}$ in partial fractions. Hence evaluate $\displaystyle \int_{1}^{3} \dfrac{2x^{2}+3x}{x^{2}+3x+2}\,dx$ …
7It is given that $f(x)$ is such that $f'(x) = \sin 2x + \cos 3x$ . Given also that $f\left(\frac{\pi}{6}\right) = 0$ , sho…
8Differentiate $\ln\left(\frac{3x}{x^2+1}\right)$ with respect to $x$ . Hence find $\displaystyle \int \frac{2x}{x^2+1}\,dx$ …
9A curve has an equation $y = 2x\sqrt{2x+1}$ . Show that $\frac{dy}{dx} = \frac{ax+b}{\sqrt{2x+1}}$ , where $a$ and $b$ are…
10Show that $\frac{d}{dx}(\tan^{3} 2x) = 6\sec^{4} 2x - 6\sec^{2} 2x$ . Hence, find the exact value of $\displaystyle \int_{0}^{\pi/8} \sec^{4} 2x\, dx$ …
11Differentiate $4xe^{3x}$ with respect to $x$ . Hence, evaluate $\int_{0}^{3} 6xe^{3x} \text{d}x$ , giving your answer in e…
12Differentiate $3x \sin 2x$ with respect to $x$ . Hence, evaluate $\int_{0}^{\frac{\pi}{4}} x \cos 2x \text{ d}x$ .
13Given that $y = 2\cos x \sin 2x - \sin x \cos 2x$ , show that $\dfrac{dy}{dx} = k\cos x \cos 2x$ , stating the value of $k$ …
14Answer all the questions A curve has an equation $y = 2x\sqrt{2x+1}$ . Show that $\frac{dy}{dx} = \frac{ax+b}{\sqrt{2x+1}}$ …
15The gradient function of a curve is given by $\frac{\text{d}y}{\text{d}x} = \frac{16}{x^3} - 4$ . Given that the line $2y - x - 6 = 0$ …
16It is given that $\mathrm{f}(x)$ is such that $\mathrm{f}'(x) = 3 \sin x \cos x$ and $\left(\frac{\pi}{2}, 1\right)$ is…
17A curve is such that $\frac{d^2y}{dx^2} = 6e^{3x} - x$ and passes through the point $\left(2, \frac{2}{3}e^6\right)$ . Gi…
18Find the derivative of the functions $y = \ln x$ and $y = \ln 2x$ respectively. Explain why, in (i), having the same der…
19Show that $\dfrac{d}{dx}(\cot x) = -\csc^{2} x$ . Hence find the value of $\displaystyle \int_{\pi/4}^{\pi/2} -\cot^{2} x\, dx$ …
20A cyclist moves along a straight road from home $O$ . His velocity, $v\text{ m/min}$ , is given by $v = 25 + 10t - 3t^2$ ,…
21Find $\frac{\text{d}}{\text{d}x}(x \cos 2x)$ . Find $\frac{\text{d}}{\text{d}x}(x^2 \sin 2x)$ . Using the results to parts…
22It is given that $y = (x-3)\sqrt{x+1}$ . Find $\dfrac{dy}{dx}$ in the form $\dfrac{ax-b}{2\sqrt{x+1}}$ , where $a$ and $b$ …
23Solutions to this question by accurate drawing will not be accepted. $ABCD$ is a trapezium where $AD$ is parallel to…
24The diagram shows part of the curve $y = \frac{1}{4}x^2 - \frac{1}{2}\ln\left(\frac{x}{3}\right)$ and $P(3,\,2.25)$ is…
25Given that $y = e^x (1+\cos x)$ , show $\dfrac{dy}{dx} = e^x (1+\cos x - \sin x)$ . Hence find $\displaystyle \int e^x (\cos x - \sin x)\,dx$ …
26Given that $y = \dfrac{4x}{\sqrt{3-2x}}$ , show that $\dfrac{dy}{dx} = \dfrac{12-4x}{\sqrt{(3-2x)^3}}$ . Hence find the va…
27Express $\dfrac{8 - 3x}{(2x + 3)(x^2 + 4)}$ in partial fractions. Differentiate $\ln\left(x^2 + 4\right)$ with respect t…
28The gradient function of a curve is $3\cos 4x - \sec^2 3x$ . Given that the curve passes through the point $\left(\frac{\pi}{12}, \frac{\sqrt{3}}{4} - \frac{1}{3}\right)$ …
29A particular curve for which $\frac{d^2y}{dx^2} = \sin 3x - \cos \frac{1}{2}x$ passes through the point $Q\left(\frac{\pi}{3}, 2\sqrt{3}\right)$ …
30Find $\dfrac{\mathrm{d}}{\mathrm{d}x}\left(3x^{2}\ln x\right). Hence find$ \int x\ln x,\mathrm{d}x $. The diagram shows the line$ …
31Show that $\frac{d}{dx} \left[ (2x)(3-5x)^{\frac{6}{5}} \right] = (6-22x)(3-5x)^{\frac{1}{5}}$ . Hence, evaluate $\int x(3-5x)^{\frac{1}{5}} dx$ …
32A curve is such that $\mathrm{f}''(x) = 6x - 10$ . The tangent to the curve $y = \mathrm{f}(x)$ at the point $(1, -2)$ pa…
33A curve is such that $\frac{d^2y}{dx^2} = 36e^{3x} - e^{-x}$ . The curve intersects the $y$ -axis at $Q(0, 5)$ and the tan…
34Given that $y = x^2 \sqrt{2x-1}$ , show that $\frac{\text{d}y}{\text{d}x} = \frac{x(5x-2)}{\sqrt{2x-1}}$ . Determine wheth…
35Find $\dfrac{\mathrm{d}}{\mathrm{d}x}\bigl((2x-3)e^{2x}\bigr). Hence evaluate$ \displaystyle\int_0^1 4xe^{2x},\mathrm{d…
36Differentiate $2x \cos 3x$ with respect to $x$ . Hence, find $\int x \sin 3x \text{ d}x$ .
37Differentiate $x \sin x + \cos x$ with respect to $x$ . Show that $\frac{\text{d}}{\text{d}x} \left( \frac{1}{3} x \sin^3 x \right) = \frac{1}{3} \sin^3 x - x \cos^3 x + x \cos x$ …
38Given that $y = (x + 2)\sqrt{x - 1}$ , show that $\frac{\text{d}y}{\text{d}x} = \frac{kx}{2\sqrt{x - 1}}$ where $k$ is co…
39Integrate $\frac{4}{x^2} - \frac{3}{1 - 5x} + \sin 5x - \pi$ with respect to $x$ .
40The equation of a curve is $y = \frac{1 - \sin x}{\cos x}$ . Show that $\frac{dy}{dx} = \tan x \sec x - \sec^2 x$ . Hence…
41A curve is such that $\dfrac{d^2y}{dx^2} = 4 - 3x$ and the point $P(2,5)$ lies on the curve. The gradient of the curve a…
42For a particular curve $\frac{\text{d}^2 y}{\text{d}x^2} = 3x - 2$ . The tangent to the curve at the point $A(4, -40)$ is…
43Express $\frac{2x^3 + x^2 + x}{(x-1)(x^2 + 1)}$ in partial fractions. Differentiate $\ln(x^2 + 1)$ with respect to $x$ .…
44Given that $y = (2x - 1)^3 \ln \sqrt{2x - 1}$ . Show that $\frac{\text{d}y}{\text{d}x} = 6(2x - 1)^2 \ln \sqrt{2x - 1} + (2x - 1)^2$ …
45 $f(x)$ is such that $f''(x) = 18x - 4$ . Given that $f(0) = 1$ and $f(2) = 9$ , find $f(x)$ . Show that $x + 1$ is a factor…
46A particle $X$ travels in a straight line so that $t$ seconds after leaving a fixed point $O$ , its velocity $v$ m/s, is…
47It is given that $f'(x) = \sin 3x - \frac{1}{2x + 1}$ and $f(0) = \frac{2}{3}$ . Find an expression for $6f(x) + f''(x)$ .
48Show that $\frac{d}{dx}\left(\frac{\ln x}{x^4}\right) = \frac{1}{x^5} - \frac{4\ln x}{x^5}$ . Given that the curve $y = \mathrm{f}(x)$ …
49The gradient function of a curve is given by $\frac{dy}{dx} = -\frac{1}{x^2} + 6 \cos 3x$ . The coordinates of the point…
50Answer all questions. Differentiate $2xe^{1-2x}$ with respect to $x$ . Hence find the exact value of $\int_{0}^{1} 2xe^{1-2x} \text{d}x$ …
51Show that $\frac{d}{dx}\left[x(3x-2)^{\frac{4}{3}}\right] = (7x-2)(3x-2)^{\frac{1}{3}}$ . Hence evaluate $\int x(3x-2)^{\frac{1}{3}} dx$ …
52Express $\frac{10x}{3-5x}$ in the form $a + \frac{b}{3-5x}$ , where $a$ and $b$ are integers. Hence, integrate $\frac{10x}{3-5x}$ …
53A curve $y = \text{f}(x)$ is such that $\text{f}''(x) = 12\text{e}^{3x} + 7$ and the point $A\left(0, -\frac{2}{3}\right)$ …
54A curve is such that $\frac{dy}{dx} = \frac{2}{x^2} + \frac{3}{(7 - 2x)}$ and $P(3, 1)$ is a point on the curve. Find th…
55Differentiate $2x \cos 2x$ with respect to $x$ . Hence, evaluate $\int_{0}^{\frac{\pi}{2}} (1 - 4x \sin 2x) \, dx$ .
56A curve is such that $\frac{\text{d}^2y}{\text{d}x^2} = \frac{12}{(3-2x)^2}$ . Given that the curve passes through the po…
57Differentiate $\ln\left(1+3x^2\right)^2$ with respect to $x$ . Hence, find $\displaystyle \int \frac{6x}{1+3x^2}\, dx$ .
58Show that $\frac{\text{d}}{\text{d}x}(2x \ln x) = 2 \ln x + 2$ . A curve is such that the gradient of its tangent is $\ln x$ …
59A curve is such that $\dfrac{d^2y}{dx^2}=3e^{2x}-e^{-x}$ . The curve cuts the y-axis at P (0, 1) and has a gradient of 4…
6013 Find $\displaystyle\int\left(\frac{3}{\sqrt{2+7x}}+\sin 5x\right)\,dx.$ If $f'(x)=2x$ and $g'(x)=2x$ , can we conclude…