Math topic中文

Standard integrals

integration of $x^{n}$ for any rational $n$ , $\sin x$ , $\cos x$ , $\sec^{2}x$ and $e^{x}$ , together with constant multiples, sums and differences

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 figure shows part of the curve $y = g(x)$ . $(2, 8)$ and $(6, 1)$ are two points on the curve. Given that $\int_{2}^{6} y \text{ } dx = 32$ …
3Express $\dfrac{x^2+3x+2}{x^2(2x+1)}$ in partial fractions. Hence integrate $\dfrac{x^2+3x+2}{x^2(2x+1)}$ with respect t…
4A particle starts from rest from a point $O$ and moves in a straight line such that its velocity $v$ m/s, is given by $v = 24t - 6t^2$ …
5The 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…
6A particle, P, starts moving in a straight line with a velocity of 8 m/s at a displacement of 3 m from a fixed point O.…
7A particle $P$ moves in a straight line so that, $t$ seconds after passing through a fixed point $O$ , its velocity $v$ m…
8Differentiate $\ln\left(\frac{3x}{x^2+1}\right)$ with respect to $x$ . Hence find $\displaystyle \int \frac{2x}{x^2+1}\,dx$ …
9Show that $\frac{d}{dx}(\tan^{3} 2x) = 6\sec^{4} 2x - 6\sec^{2} 2x$ . Hence, find the exact value of $\displaystyle \int_{0}^{\pi/8} \sec^{4} 2x\, dx$ …
10Given 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 $k$ …
11 $\int _{2}^{4}\dfrac{{{x}^{3}}-3{{x}^{2}}+5}{{{x}^{2}}}dx$ 的值为 $()$ A. $1$ B. $\dfrac{1}{4}$ C. $\dfrac{5}{4}$ …
12Answer all the questions A curve has an equation $y = 2x\sqrt{2x+1}$ . Show that $\frac{dy}{dx} = \frac{ax+b}{\sqrt{2x+1}}$ …
13The gradient function of a curve is given by $\frac{\text{d}y}{\text{d}x} = \frac{16}{x^3} - 4$ . Given that the line $2y - x - 6 = 0$ …
14计算定积分 $\displaystyle\int_0^2 (3x^2 + 2x) \, dx$ 。
15A particle P moves along a horizontal straight line so that t seconds after motion has begun, its displacement, s m, fro…
16A 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…
17A particle moves in a straight line so that its velocity, $v$ m/s is given by $v = t^2 - 4t + 3$ , where $t$ is the time…
18A particle travels in a straight line so that, $t$ seconds after leaving a fixed point $O$ , its velocity, $v\text{ m s}^{-1}$ …
19Find the area of the shaded region. The figure above is not drawn to scale.
20Show that $\dfrac{d}{dx}(\cot x) = -\csc^{2} x$ . Hence find the value of $\displaystyle \int_{\pi/4}^{\pi/2} -\cot^{2} x\, dx$ …
21A cyclist moves along a straight road from home $O$ . His velocity, $v\text{ m/min}$ , is given by $v = 25 + 10t - 3t^2$ ,…
22The diagram shows part of the curve $y = x^3 + 3x^2 - 2$ . The straight line $L$ cuts the curve at $A(-3, -2)$ , the x-axi…
23Find $\frac{\text{d}}{\text{d}x}(x \cos 2x)$ . Find $\frac{\text{d}}{\text{d}x}(x^2 \sin 2x)$ . Using the results to parts…
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 $\displaystyle \int e^x (\cos x - \sin x)\,dx$ …
26The diagram shows the curve $y = \cos x + \sin x$ for $0 \le x \le \pi$ . The curve meets the $y$ -axis at the point $A$ a…
27The diagram shows part of the curve $y = \frac{15}{(0.5x+1)}$ and a straight line $y = 1.25x - 10$ . The straight line in…
28Find $\dfrac{\mathrm{d}}{\mathrm{d}x}\left(3x^{2}\ln x\right). Hence find$ \int x\ln x,\mathrm{d}x $. The diagram shows the line$ …
29The diagram shows part of the curve $y = \frac{5}{2x - 1}$ . The line $9y = -10x + 55$ intersects the curve at points $A$ …
30A curve is such that $\mathrm{f}''(x) = 6x - 10$ . The tangent to the curve $y = \mathrm{f}(x)$ at the point $(1, -2)$ pa…
31A particle $P$ travels in a straight line and its velocity, $v$ m/s, is given by $v = 4\sin^2\left(\frac{t}{2}\right) - 1$ …
32Express $\frac{13x - 6}{x^2(2x - 3)}$ in partial fractions. Hence $\int \frac{13x - 6}{x^2(2x - 3)} \, dx$ .
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…
34A particle moving in a straight line, passes through a fixed point $O$ with a velocity of $4\text{ m/s}$ . Its accelerati…
35Find $\dfrac{\mathrm{d}}{\mathrm{d}x}\bigl((2x-3)e^{2x}\bigr). Hence evaluate$ \displaystyle\int_0^1 4xe^{2x},\mathrm{d…
36Differentiate $x \sin x + \cos x$ with respect to $x$ . Show that $\frac{\text{d}}{\text{d}x} \left( \frac{1}{3} x \sin^3 x \right) = \frac{1}{3} \sin^3 x - x \cos^3 x + x \cos x$ …
37Given 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…
38Integrate $\frac{4}{x^2} - \frac{3}{1 - 5x} + \sin 5x - \pi$ with respect to $x$ .
39The 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…
40A 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…
41Express $\frac{3x^2 - 4}{x^2(3x - 2)}$ in partial fractions. Hence, evaluate $\int \frac{3x^2 - 4}{x^2(3x - 2)} \, dx$ .
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…
43The diagram shows part of the curve $y=(9-x)(x-3)$ and the line $y=k-3$ , where $k>3$ . The line through the maximum point…
44Integrate $\left(x - \frac{5}{2x}\right)^2$ with respect to $x$ .
45Express $\frac{2x^3 + x^2 + x}{(x-1)(x^2 + 1)}$ in partial fractions. Differentiate $\ln(x^2 + 1)$ with respect to $x$ .…
46 $f(x)$ 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…
47Integrate $\left(\frac{6}{e^{3-x}}\right)^2$ with respect to $x$ . Explain, with working, why the equation $2x^3 + 3x^2 + 2x + 8 = 0$ …
48A particle $X$ travels in a straight line so that $t$ seconds after leaving a fixed point $O$ , its velocity $v$ m/s, is…
49Show 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 $y = \mathrm{f}(x)$ …
50The diagram shows the curve $y = 4 - 2\cos 2x$ for $0 \le x \le \frac{3}{2}\pi$ radians and a straight line passing thro…
51The gradient function of a curve is given by $\frac{dy}{dx} = -\frac{1}{x^2} + 6 \cos 3x$ . The coordinates of the point…
52A particle moves in a straight line, so that $t$ seconds after passing through a fixed point $O$ , its acceleration, $a\text{ ms}^{-2}$ …
53Show that $\frac{d}{dx}\left[x(3x-2)^{\frac{4}{3}}\right] = (7x-2)(3x-2)^{\frac{1}{3}}$ . Hence evaluate $\int x(3x-2)^{\frac{1}{3}} dx$ …
54Integrate $\frac{8}{2x-1} + \frac{4}{x^3} + 1$ with respect to $x$ .
55Express $\frac{10x}{3-5x}$ in the form $a + \frac{b}{3-5x}$ , where $a$ and $b$ are integers. Hence, integrate $\frac{10x}{3-5x}$ …
56By using long division, divide $4x^3 + 5x^2 + x - 1$ by $x^2 (x+1)$ . Hence, express $\dfrac{4x^3 + 5x^2 + x - 1}{x^2 (x+1)}$ …
57A particle travels in a straight line, so that $t$ seconds after passing a fixed point $O$ , its velocity, $v$ m/s, is gi…
58The diagram shows part of the curve of $y = x(x-2)^2$ , which passes through the point $P(1, 1)$ . Show that the equation…
59A curve $y = \text{f}(x)$ is such that $\text{f}''(x) = 12\text{e}^{3x} + 7$ and the point $A\left(0, -\frac{2}{3}\right)$ …
60A 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…