$\int e^{x^4}\left(x+x^3+2 x^5\right) e^{x^2} d x$ is equal to
Answer & explanation
Correct answer: option 4
Let $x^2=t$. Then,
$I=\int e^{x^4}\left(x+x^3+2 x^5\right) e^{x^2} d x=\frac{1}{2} \int e^{t^2}\left(1+t+2 t^2\right) e^t d t$
$\Rightarrow I=\frac{1}{2} \int e^t\left\{t e^{t^2}+\left(e^{t^2}+2 t^2 e^{t^2}\right)\right\} d t$
$\Rightarrow I=\frac{1}{2} e^t\left(t e^{t^2}\right)+C=\frac{1}{2} x^2 e^{x^2} e^{x^4}+C$