Question

Evaluate the integral

${\int }_0^{\pi }\frac{xdx}{4si{n}^{2}x+5}$

• A. $\frac{{\pi }^2}{6\sqrt{5}}$
• B. $\frac{{\pi }^2}{20}$
• C. $\frac{{\pi }^2}{9}$
• D. $\frac{{\pi }^2}{3\sqrt{5}}$

Answer 1

The correct option is A $\frac{{\pi }^2}{6\sqrt{5}}$

$I={\int }_0^{\pi }\frac{xdx}{4si{n}^{2}x+5}$ ---- (1)

$I={\int }_0^{\pi }\frac{(\pi -x)dx}{4si{n}^{2}x+5}$ ---- (2)

$\to 2I=\pi {\int }_0^{\pi }\frac{dx}{4si{n}^{2}x+5}$

$2I=\pi {\int }_0^{\pi }\frac{dtanx}{4ta{n}^{2}x+5(ta{n}^{2}x+1)}$

$=\pi {\int }_0^{\pi }\frac{dtanx}{9ta{n}^{2}x+5}$

$=\frac{\pi }{\sqrt{5}\times 3}\left[ta{n}^{-1}\right(tanx\left)\right]{|}_0^{\pi }$

$2I=\frac{{\pi }^2}{3\sqrt{5}}$

$I=\frac{{\pi }^2}{6\sqrt{5}}$