Math topic中文

Integration linear argument

integration of $(ax + b)^{n}$ for any rational $n$ , $\sin(ax + b)$ , $\cos(ax + b)$ and $e^{(ax + b)}$

Practice questions

1The diagram shows part of the curve $y = 3e^{-\frac{1}{2}x} - 6$ which crosses the $x$ -axis at $P$ and the $y$ -axis at $Q$ …
2Express $\dfrac{x^2+3x+2}{x^2(2x+1)}$ in partial fractions. Hence integrate $\dfrac{x^2+3x+2}{x^2(2x+1)}$ with respect t…
3A motorist, travelling at a constant velocity of V m/s, passed a fixed point X and saw few vehicles ahead. He stepped on…
4Express $\frac{8x+13}{(1+2x)(2+x)^2}$ in partial fractions. Express $\frac{8x+13}{(1+2x)(2+x)^2}$ in partial fractions…
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…
6The diagram shows part of the curve $y = \sqrt{3x + 4}$ . The line $AB$ is tangent to the curve at $A$ . The points $B$ an…
7It 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 $f''\left(\frac{\pi}{12}\right)$ …
8The 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…
9It is given that $y = 4x \sin 2x$ . Show that $\frac{\mathrm{d}y}{\mathrm{d}x}$ can be expressed in the form $ax \cos 2x + b \sin 2x$ …
10Express $\dfrac{2x^{2}+3x}{x^{2}+3x+2}$ in partial fractions. Hence evaluate $\displaystyle \int_{1}^{3} \dfrac{2x^{2}+3x}{x^{2}+3x+2}\,dx$ …
11It 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…
12A particle moves in a straight line, so that, $t$ seconds after leaving a fixed point $O$ its velocity, $v$ is given by…
13The diagram below shows the graph of $y = \frac{1}{2x+1}$ and the tangent to the curve where the curve intersects the $y$ …
14Show that $\frac{d}{dx}(\tan^{3} 2x) = 6\sec^{4} 2x - 6\sec^{2} 2x$ . Hence, find the exact value of $\displaystyle \int_{0}^{\pi/8} \sec^{4} 2x\, dx$ …
15Express $\frac{2x+15}{(3+x)(3-2x)}$ in partial fractions. Hence evaluate $\int_{0}^{1} \frac{2x+15}{(3+x)(3-2x)} \, dx$ .
16The 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…
17Differentiate $4xe^{3x}$ with respect to $x$ . Hence, evaluate $\int_{0}^{3} 6xe^{3x} \text{d}x$ , giving your answer in e…
18Differentiate $3x \sin 2x$ with respect to $x$ . Hence, evaluate $\int_{0}^{\frac{\pi}{4}} x \cos 2x \text{ d}x$ .
19A particle passes a fixed point $O$ and moves in a straight line such that, $t$ s after leaving $O$ , its velocity, $v\text{ m/s}$ …
20A particle moves in a straight line such that its velocity, $v\text{ m/s}$ , is given by $v = 9\sin\left(\frac{t}{2}\right)$ …
21The diagram shows part of the curve $y = 4 - e^{-2x}$ which crosses the axes at $A$ and at $B$ . Find the coordinates of…
22The 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…
23Evaluate $\int_{0}^{\frac{\pi}{3}} \left( \frac{1}{2} \sin 3x - \cos^2 x \right) dx$ , correct to 2 decimal places.
24It 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…
25The diagram shows part of the curve $y = \sqrt{4x + 5}$ . The normal of the curve at $P$ meets the $x$ -axis at $T$ . The $x$ …
26A 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…
27Find the area of the shaded region. The figure above is not drawn to scale.
28The 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…
29The 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…
30The 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…
31Find $\frac{\text{d}}{\text{d}x}(x \cos 2x)$ . Find $\frac{\text{d}}{\text{d}x}(x^2 \sin 2x)$ . Using the results to parts…
32It 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 $b$ …
33The diagram shows part of the curve $y = 2\sin\frac{x}{2}$ , where $A$ lies on the curve. The $x$ -coordinate of $A$ is $\frac{4\pi}{3}$ …
34Express $\dfrac{8 - 3x}{(2x + 3)(x^2 + 4)}$ in partial fractions. Differentiate $\ln\left(x^2 + 4\right)$ with respect t…
35The gradient function of a curve is $3\cos 4x - \sec^2 3x$ . Given that the curve passes through the point $\left(\frac{\pi}{12}, \frac{\sqrt{3}}{4} - \frac{1}{3}\right)$ …
36The diagram shows part of the curve $y = \frac{15}{(0.5x+1)}$ and a straight line $y = 1.25x - 10$ . The straight line in…
37A particular curve for which $\frac{d^2y}{dx^2} = \sin 3x - \cos \frac{1}{2}x$ passes through the point $Q\left(\frac{\pi}{3}, 2\sqrt{3}\right)$ …
38Show that $\frac{d}{dx} \left[ (2x)(3-5x)^{\frac{6}{5}} \right] = (6-22x)(3-5x)^{\frac{1}{5}}$ . Hence, evaluate $\int x(3-5x)^{\frac{1}{5}} dx$ …
39The diagram shows part of the curve $y = \frac{5}{2x - 1}$ . The line $9y = -10x + 55$ intersects the curve at points $A$ …
40The diagram shows part of the curve $y = \frac{7}{6x + 1}$ intersecting the $y$ -axis at $A(0, 7)$ . The tangent to the cu…
41A particle moves in a straight line such that, $t$ s after passing through a fixed point $O$ , its displacement from $O$ …
42By using a suitable substitution, or otherwise, solve the equation $3\log_6 x = \log_x 3 + \log_x 2 + \log_5 25.$ Sket…
43Given that $y = \sec x$ , show that $\frac{dy}{dx} = \sec x \tan x$ . Hence show that $y^4 - y^2 = \left(\frac{dy}{dx}\right)^2$ …
44Express $\frac{13x - 6}{x^2(2x - 3)}$ in partial fractions. Hence $\int \frac{13x - 6}{x^2(2x - 3)} \, dx$ .
45A 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…
46Given 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…
47A particle $P$ moves in a straight line, so that, $t$ seconds after leaving a fixed point $O$ , its velocity, $v$ m/s, is…
48A particle moves from point P towards point O. The speed of the particle, $t$ seconds after passing point P, can be mode…
49Differentiate $2x \cos 3x$ with respect to $x$ . Hence, find $\int x \sin 3x \text{ d}x$ .
50Integrate $\frac{4}{x^2} - \frac{3}{1 - 5x} + \sin 5x - \pi$ with respect to $x$ .
51The 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…
52Express $\frac{3x^2 - 4}{x^2(3x - 2)}$ in partial fractions. Hence, evaluate $\int \frac{3x^2 - 4}{x^2(3x - 2)} \, dx$ .
53A scooter travelling up and down a straight horizontal road passes a fixed point O with a speed of $p\ \text{m s}^{-1}$ .…
54Express $\frac{9x^2 + 17x - 16}{3x^2 + 5x - 2}$ in partial fractions. Hence, find $\int \frac{9x^2 + 17x - 16}{3x^2 + 5x - 2} \, dx$ …
55Express $\frac{2x^3 + x^2 + x}{(x-1)(x^2 + 1)}$ in partial fractions. Differentiate $\ln(x^2 + 1)$ with respect to $x$ .…
56A particle moves in a straight line with a fixed point $O$ . The velocity of the particle, $v$ m/s, is given by $v = \dfrac{25}{2t+1} + 10$ …
57Given that $y = (2x - 1)^3 \ln \sqrt{2x - 1}$ . Show that $\frac{\text{d}y}{\text{d}x} = 6(2x - 1)^2 \ln \sqrt{2x - 1} + (2x - 1)^2$ …
58Integrate $\left(\frac{6}{e^{3-x}}\right)^2$ with respect to $x$ . Explain, with working, why the equation $2x^3 + 3x^2 + 2x + 8 = 0$ …
59The 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…
60It is given that $f'(x) = \sin 3x - \frac{1}{2x + 1}$ and $f(0) = \frac{2}{3}$ . Find an expression for $6f(x) + f''(x)$ .