Question

Evaluate the definite integral: $\int_{0}^{\frac{\pi}{4}}(\frac{sinxcosx}{cos^{4}x+sin^{4}x})dx$

Answer 1

$Let \space I =\int_{0}^{\frac{\pi}{4}}(\frac{sinxcosx}{cos^{4}x+sin^{4}x})dx$

$⇒I=\int_{0}^{\frac{\pi}{4}}\frac{\frac{(sinxcosx)}{cos^{4}x}}{\frac{(cos^{4}x+sin^{4}x)}{cos^{4}x}}dx$

$⇒I=\int_{0}^{\frac{\pi}{4}}\frac{tanxsec^{2}x}{1+tan^{4}x}dx$

$Let \space tan^{2}x=t⇒2tanxsec^{2}xdx=dt$

$When x=0,t=0 \space and \space when \space x=\frac{\pi}{4},t=1$

$∴I=\frac{1}{2}\int_{0}^{1}\frac{dt}{1+t^{2}}$

$=\frac{1}{2}[tan^{-1}t]_{0}^{1}$

$=\frac{1}{2}[tan^{-1}1-tan^{-1}0]$

$=\frac{1}{2}[\frac{\pi}{4}]$

$=\frac{\pi}{8}$