Question:
Evaluate $\int x \sin ^{5} x^{2} \cos x^{2} d x$
Solution:
Let, $\sin x^{2}=t$
Differentiating both sides with respect to $x$
$\frac{d t}{d x}=\cos x^{2} \times 2 x \Rightarrow \frac{d t}{2}=x \cos x^{2} d x$
$y=\int \frac{t^{5}}{2} d t$
Using formula $\int t^{n} d t=\frac{t^{n+1}}{n+1}$
$y=\frac{t^{6}}{2 \times 6}+c$
Again, put $t=\sin x^{2}$
$y=\frac{\sin ^{6} x^{2}}{12}+c$