Question

Evaluate: $\int x^2\frac{d}{dx}\left(e^{x^2}\right)dx$

• A. $e^{x^2}(x-1)+c$
• B. $e^{x^2}(x^2-1)+c$
• C. $\frac{1}{2}{e}^{x^2}(x-1)+c$
• D. $e^{x^3}(x^2-1)+c$

Answer 1

Answer: (B)

$e^{x^2}(x^2-1)+c$

$\int x^2\frac{d}{dx}\left(e^{x^2}\right)dx=\int 2x^3{e}^{x^2}dx$

Put $x^2=t$ , we have $2xdx=dt$

$\Rightarrow \int 2x^3{e}^{x^2}dx=\int t{e}^{t}dt$

By applying integration byparts , we get $\int t{e}^{t}dt=t{e}^t-\int e^{t}dt+c=e^t(t-1)+c$

Now substitute $t=x^2$

We get $e^{x^2}(x^2-1)+c$

Therefore option $B$ is correct