Question:
Evaluate the following integrals:
$\int x^{3} \sin \left(x^{4}+1\right) d x$
Solution:
Assume $x^{4}+1=t$
$d\left(x^{4}+1\right)=d t$
$4 x^{3} d x=d t$
$x^{3} d x=\frac{d t}{4}$
Substituting $\mathrm{t}$ and $\mathrm{dt}$
$\Rightarrow \int \frac{1}{4} \sin t d t$
$\Rightarrow \frac{-1 \cos t}{4}+c$
But $t=x^{4}+1$
$\Rightarrow \frac{-1}{4} \cos \left(x^{4}+1\right)+c$
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