Math topic中文

Evaluating definite integrals

evaluation of definite integrals

Practice questions

1The figure shows part of the curve $y = g(x)$ . $(2, 8)$ and $(6, 1)$ are two points on the curve. Given that $\int_{2}^{6} y \text{ } dx = 32$ …
2A motorist, travelling at a constant velocity of V m/s, passed a fixed point X and saw few vehicles ahead. He stepped on…
3Express $\frac{8x+13}{(1+2x)(2+x)^2}$ in partial fractions. Express $\frac{8x+13}{(1+2x)(2+x)^2}$ in partial fractions…
4It is given that $y = 4x \sin 2x$ . Show that $\frac{\mathrm{d}y}{\mathrm{d}x}$ can be expressed in the form $ax \cos 2x + b \sin 2x$ …
5Express $\dfrac{2x^{2}+3x}{x^{2}+3x+2}$ in partial fractions. Hence evaluate $\displaystyle \int_{1}^{3} \dfrac{2x^{2}+3x}{x^{2}+3x+2}\,dx$ …
6The diagram shows part of the curve $y = (1 - x)(x - 4)$ and two lines $l_1$ and $l_2$ . The lines $l_1$ and $l_2$ are ta…
7A particle moves in a straight line, so that, $t$ seconds after leaving a fixed point $O$ its velocity, $v$ is given by…
8The diagram below shows the graph of $y = \frac{1}{2x+1}$ and the tangent to the curve where the curve intersects the $y$ …
9Show that $\frac{d}{dx}(\tan^{3} 2x) = 6\sec^{4} 2x - 6\sec^{2} 2x$ . Hence, find the exact value of $\displaystyle \int_{0}^{\pi/8} \sec^{4} 2x\, dx$ …
10Express $\frac{2x+15}{(3+x)(3-2x)}$ in partial fractions. Hence evaluate $\int_{0}^{1} \frac{2x+15}{(3+x)(3-2x)} \, dx$ .
11The diagram shows part of the curve $y = \frac{1}{\left(\frac{3}{4}x - 5\right)^2} + 1$ . $P$ is the point on the curve w…
12Differentiate $4xe^{3x}$ with respect to $x$ . Hence, evaluate $\int_{0}^{3} 6xe^{3x} \text{d}x$ , giving your answer in e…
13Differentiate $3x \sin 2x$ with respect to $x$ . Hence, evaluate $\int_{0}^{\frac{\pi}{4}} x \cos 2x \text{ d}x$ .
14 $∫_{-2}^{2}(\sin x+\sqrt{4-x^{2}}){\rm d}x=(\quad)$ A. $4$ B. $2π$ C. $4+2π$ D. $8$
15Given 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 $k$ …
16 $\int _{2}^{4}\dfrac{{{x}^{3}}-3{{x}^{2}}+5}{{{x}^{2}}}dx$ 的值为 $()$ A. $1$ B. $\dfrac{1}{4}$ C. $\dfrac{5}{4}$ …
17A particle moves in a straight line such that its velocity, $v\text{ m/s}$ , is given by $v = 9\sin\left(\frac{t}{2}\right)$ …
18The diagram shows part of the curve $y = 4 - e^{-2x}$ which crosses the axes at $A$ and at $B$ . Find the coordinates of…
19The diagram shows part of the curve $y = e^{x-1} - 2$ . The point $P(2,\ e-2)$ lies on the curve and the normal to the cu…
20计算定积分 $\displaystyle\int_0^2 (3x^2 + 2x) \, dx$ 。
21Evaluate $\int_{0}^{\frac{\pi}{3}} \left( \frac{1}{2} \sin 3x - \cos^2 x \right) dx$ , correct to 2 decimal places.
22Find the area of the shaded region. The figure above is not drawn to scale.
23Show that $\dfrac{d}{dx}(\cot x) = -\csc^{2} x$ . Hence find the value of $\displaystyle \int_{\pi/4}^{\pi/2} -\cot^{2} x\, dx$ …
24The graph shows part of the curve $y = -3(2x-4)^{-1/3}$ . The point $P(-2,\ 1.5)$ lies on the curve and the normal to the…
25The diagram shows part of the curve $y = \sqrt{2x + 4}$ , meeting the $x$ -axis at the point $A$ and the $y$ -axis at the p…
26The diagram shows part of the curve $y = 2 \sin 2x + 1$ . $P$ is the maximum point of the curve and $Q$ is the point on t…
27The 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…
28It 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 $b$ …
29The 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…
30The diagram shows part of the curve $y = 2\sin\frac{x}{2}$ , where $A$ lies on the curve. The $x$ -coordinate of $A$ is $\frac{4\pi}{3}$ …
31求曲线 $y = x^2$ 与 $y = 2x$ 围成的封闭图形的面积。
32Given that $y = \dfrac{4x}{\sqrt{3-2x}}$ , show that $\dfrac{dy}{dx} = \dfrac{12-4x}{\sqrt{(3-2x)^3}}$ . Hence find the va…
33The 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…
34Express $\dfrac{8 - 3x}{(2x + 3)(x^2 + 4)}$ in partial fractions. Differentiate $\ln\left(x^2 + 4\right)$ with respect t…
35The diagram shows part of the curve $y = \frac{15}{(0.5x+1)}$ and a straight line $y = 1.25x - 10$ . The straight line in…
36Find $\dfrac{\mathrm{d}}{\mathrm{d}x}\left(3x^{2}\ln x\right). Hence find$ \int x\ln x,\mathrm{d}x $. The diagram shows the line$ …
37It is given that $f(x) = \begin{cases} \sqrt{4a^2 - x^2} & \text{for } 0 \le x \le 2a, \\ 3x - 6a & \text{for } 2a < x < 3a, \end{cases}$ …
38The diagram shows part of the curve $y = \frac{5}{2x - 1}$ . The line $9y = -10x + 55$ intersects the curve at points $A$ …
39The diagram shows part of the curve $y = \frac{7}{6x + 1}$ intersecting the $y$ -axis at $A(0, 7)$ . The tangent to the cu…
40已知 $a$ 是 $A=7^{n}+C_{n}^{1}7^{n-1}+C_{n}^{2}7^{n-2}+⋯+C_{n}^{n-1}7(n$ 为正奇数 $)$ 被 $4$ 整除的余数，则 $∫_{1}^{e}(x+\dfrac{a}{x}){\rm d}x$ …
41设 $a= \int _{ 0 }^{ π }(\sin x+\cos x)dx$ ，且二项式 $(a \sqrt {x}- \dfrac {1}{ \sqrt {x}})^{n}$ 的所有二项式系数之和为 $64$ ，则其展开式中含 $x^{2}$ …
42Given 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…
43Find $\dfrac{\mathrm{d}}{\mathrm{d}x}\bigl((2x-3)e^{2x}\bigr). Hence evaluate$ \displaystyle\int_0^1 4xe^{2x},\mathrm{d…
44A particle $P$ moves in a straight line, so that, $t$ seconds after leaving a fixed point $O$ , its velocity, $v$ m/s, is…
45A particle moves from point P towards point O. The speed of the particle, $t$ seconds after passing point P, can be mode…
46Given 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…
47The 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…
48A scooter travelling up and down a straight horizontal road passes a fixed point O with a speed of $p\ \text{m s}^{-1}$ .…
49The 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…
50A particle moves in a straight line with a fixed point $O$ . The velocity of the particle, $v$ m/s, is given by $v = \dfrac{25}{2t+1} + 10$ …
51Given that $y = (2x - 1)^3 \ln \sqrt{2x - 1}$ . Show that $\frac{\text{d}y}{\text{d}x} = 6(2x - 1)^2 \ln \sqrt{2x - 1} + (2x - 1)^2$ …
52The graph shows the curve $C$ with equation $y = x^2$ and the line $L$ with equation $y = 2x-1$ . The line $L$ is tangent…
53The diagram shows part of the curve $y = \sqrt{3x-2}$ . The normal to the curve at $B$ meets the $x$ -axis at $C$ . Given t…
54The 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…
55Answer all questions. Differentiate $2xe^{1-2x}$ with respect to $x$ . Hence find the exact value of $\int_{0}^{1} 2xe^{1-2x} \text{d}x$ …
56Show that $\frac{d}{dx}\left[x(3x-2)^{\frac{4}{3}}\right] = (7x-2)(3x-2)^{\frac{1}{3}}$ . Hence evaluate $\int x(3x-2)^{\frac{1}{3}} dx$ …
57The diagram shows part of the curve $y = \sqrt{2x+5}$ . A is a point on the curve and the x-coordinate of A is 2. The nor…
58The diagram shows part of the curve $y = 5 + \frac{4}{e^{2x}}$ intersecting the $y$ -axis at point $P$ . The tangent to th…
59A particle travels in a straight line, so that $t$ seconds after passing a fixed point $O$ , its velocity, $v$ m/s, is gi…
60The diagram shows part of the curve of $y = x(x-2)^2$ , which passes through the point $P(1, 1)$ . Show that the equation…