The value of the integral \int_{- \pi / 2}^{\pi / 2} \left(\right. x^{2} + ln \textrm{ } \frac{\pi + x}{\pi

Engineering

Mathematics

Properties of Definite Integral

Question

The value of the integral $\int_{- π / 2}^{π / 2} (x^{2} + \ln \frac{π + x}{π - x}) cosx dx$ is

$\frac{π^{2}}{2} + 4$

$\frac{π^{2}}{2} - 4$

$\frac{π^{2}}{2}$

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$Let I = \int_{- π / 2}^{π / 2} (x^{2} + \ln \frac{π + x}{π - x}) cosx dx$

$x - \sqrt{2} y = 1 - \sqrt{2}$

$Ι = 2 \int_{0}^{π / 2} \underset{I}{\underset{⏟}{x^{2}}} \underset{II}{\underset{⏟}{cosx}} dx = 2 {[x^{2} (sinx) - 2 x (- cosx) + 2 (- sinx)]}_{0}^{π / 2}$

Applying integration by parts

$I = \frac{π^{2}}{2} - 4$