Evaluate the definite integral: \(\int_{0}^{\frac{\pi}{4}}(\frac{sinxcosx}{cos^{4}x+sin^{4}x})dx\)
Solution and Explanation
\(Let \space I =\int_{0}^{\frac{\pi}{4}}(\frac{sinxcosx}{cos^{4}x+sin^{4}x})dx\)
\(⇒I=\int_{0}^{\frac{\pi}{4}}\frac{\frac{(sinxcosx)}{cos^{4}x}}{\frac{(cos^{4}x+sin^{4}x)}{cos^{4}x}}dx\)
\(⇒I=\int_{0}^{\frac{\pi}{4}}\frac{tanxsec^{2}x}{1+tan^{4}x}dx\)
\(Let \space tan^{2}x=t⇒2tanxsec^{2}xdx=dt\)
\(When x=0,t=0 \space and \space when \space x=\frac{\pi}{4},t=1\)
\(∴I=\frac{1}{2}\int_{0}^{1}\frac{dt}{1+t^{2}}\)
\(=\frac{1}{2}[tan^{-1}t]_{0}^{1}\)
\(=\frac{1}{2}[tan^{-1}1-tan^{-1}0]\)
\(=\frac{1}{2}[\frac{\pi}{4}]\)
\(=\frac{\pi}{8}\)
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