Question:
$\int x^{2} e^{x^{3}} d x$ equals
(A) $\frac{1}{3} e^{x^{2}}+\mathrm{C}$
(B) $\frac{1}{3} e^{x^{2}}+\mathrm{C}$
(C) $\frac{1}{2} e^{x^{3}}+\mathrm{C}$
(D) $\frac{1}{3} e^{x^{2}}+\mathrm{C}$
Solution:
Let $I=\int x^{2} e^{x^{3}} d x$
Also, let $x^{3}=t \Rightarrow 3 x^{2} d x=d t$
$\begin{aligned} \Rightarrow I &=\frac{1}{3} \int e^{t} d t \\ &=\frac{1}{3}\left(e^{t}\right)+\mathrm{C} \\ &=\frac{1}{3} e^{x^{3}}+\mathrm{C} \end{aligned}$
Hence, the correct answer is A.