Question

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

Answer 1

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

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

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

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

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

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

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

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

\(=\frac{1}{2}[\frac{\pi}{4}]\)

\(=\frac{\pi}{8}\)