Mathematics - Indefinite Integration Question with Solution | TestHub

We have, $I=\int x^5{e}^{-4x^3}dx$ 

Let, $-4x^3=t\Rightarrow -12x^{2}dx=dt$

$\Rightarrow I=\int \left(-\frac{t}{4}\right)e^t\left(-\frac{dt}{12}\right)=\frac{1}{48}\int t{e}^{t}dt$

$=\frac{1}{48}\left[t{e}^t-\int e^{t}dt\right]=\frac{\left(t-1\right)e^t}{48}+C=\frac{\left(-4x^3-1\right)e^{-4x^3}}{48}+C,$
where $C$ is the constant of integration.

Hence, $f\left(x\right)=-4x^3-1$