First slide

Evaluation of definite integrals

Question

If the value of $\int_{- 2}^{2} | x \cos π x | d x = k / π$ then the value of $k$ is

Moderate

A

4
B

8
C

12
D

none of these

Solution

Since the integrand is an even function so required integral is equal to $2 \int_{0}^{2} | x \cos π x | d x$
$\begin{aligned} = 2 [\int_{0}^{1 / 2} x \cos π x d x - \int_{1 / 2}^{3 / 2} x \cos π x d x + \int_{3 / 2}^{2} x \cos π x d x] \\ = 2 [x \frac{\sin π x}{π} + {\frac{\cos π x}{π^{2}}|}_{0}^{1 / 2} - {(x \frac{\sin π x}{π} + \frac{\cos π x}{π^{2}})|}_{1 / 2}^{3 / 2} \\ + {(\frac{x \sin π x}{π} + \frac{\cos π x}{π^{2}})|}_{y_{2}}^{2}] \end{aligned}$
$= 2 [\frac{1}{2 π} - \frac{1}{π^{2}} - (- \frac{3}{2 π} - \frac{1}{2 π}) + \frac{1}{π^{2}} - (- \frac{3}{2 π})] = \frac{8}{π}$

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