(a) Evaluate: ∫_0^π / 4 (sin x cos x)/(cos^4 x+ sin^4 x) d x

\[I=\int_{0}^{\pi / 4} \frac{\sin x \cos x}{\cos ^{4} x+\sin ^{4} x} d x \] dividing numerator and denominator by $\displaystyle \cos ^{4} x$, \[I=\int_{0}^{\pi / 4} \frac{\tan x \sec ^{2} x}{1+\tan ^{4} x} d x \] Put $\displaystyle \tan ^{2} x=t \Rightarrow 2 \tan x \sec ^{2} x d x=d t$ when $\displaystyle x=0, t=0$; when $\displaystyle x=\frac{\pi}{4}, t=1$ \[\Rightarrow I=\frac{1}{2} \int_{0}^{1} \frac{d t}{1+t^{2}} \] \[=\frac{1}{2}\left[\tan ^{-1} t\right]_{0}^{1} \] Due to printing error, the given function is not integrable. So full marks may be given for every attempt.