The problem is:
∫(x^2)e^[-(x^3)/4]
The attempt at a solution:
Is it equals to (-4/3)e^[-(x^3)/4]???
I am really confuse here. Any help would be appreciated
You actually do take the $\displaystyle x^2$ term into account!
When you make the substitution $\displaystyle u={\color{blue}-\tfrac{1}{4}x^3}$, we see that $\displaystyle \,du=-\tfrac{3}{4}{\color{red}x^2\,dx}$
At this stage, let's go back to the original integral:
$\displaystyle \int {\color{red}x^2}e^{-x^3/4}{\color{red}\,dx}$.
Observed that I highlighted the part that appeared in our $\displaystyle \,du$ term! So we see that $\displaystyle \,du=-\tfrac{3}{4}{\color{red}x^2\,dx}\implies -\tfrac{4}{3}\,du={\color{red}x^2\,dx}$.
Therefore, $\displaystyle \int {\color{red}x^2}e^{{\color{blue}-x^3/4}}{\color{red}\,dx}=\int e^u(-\tfrac{4}{3}\,du)=-\tfrac{4}{3}\int e^u\,du$.
Then integrating results in $\displaystyle -\tfrac{4}{3}e^u+C$ and back substitution gives us the desired result $\displaystyle \int x^2e^{-x^3/4}\,dx=-\tfrac{4}{3}e^{-x^3/4}+C$.
Does this clarify what's going on here?