June Revision Exercise 3 Q7

(a)

$\int x(\mathrm{ln}x)^2 ~dx$

$=\frac{x^2}{2}(\mathrm{ln}x)^2 - \int \frac{x^2}{2} 2(\mathrm{ln}x)\frac{1}{x} ~dx$

$=\frac{x^2}{2}(\mathrm{ln}x)^2 - \int x \mathrm{ln}x ~dx$

$=\frac{x^2}{2}(\mathrm{ln}x)^2 - \frac{x^2}{2} \mathrm{ln}x + \int \frac{x^2}{2}\frac{1}{x}~dx$

$=\frac{x^2}{2}(\mathrm{ln}x)^2 - \frac{x^2}{2} \mathrm{ln}x + \frac{x^2}{4} + C$

(b)

$\int \frac{1}{1-\mathrm{cos}2x} ~dx$

$= \int \frac{1}{2\mathrm{sin}^2x} ~dx$

$= \frac{2}{2} \int \mathrm{cosec}^2x~dx$

$= -\frac{1}{2} \mathrm{cot}x + C$

(c)
(i)

(ii)

$\int_{\mathrm{ln}2}^{\mathrm{ln}3} x ~dy$

$=\int_2^3 t^2 \frac{1}{t} ~dt$

$=\Big| \frac{t^2}{2} \Big|_2^3$

$= 2.5 \text{units}^2$

(iii)
$\pi \int_4^9 y^2 ~dx$

$=\pi \int_2^3 (\mathrm{ln}t)^2(2t) ~dt$

$=2\pi \int_2^3 t(\mathrm{ln}t)^2 ~dt$

$= 13.6$

Back to June Revision Exercise 3

9740 9758 A'levels Definite Integral Differentiation Exercise 3 Integration June Revision Preparation 2016 Revision

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