Find $\int x e^{(1+x^2)} dx$.
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → $\frac{1}{2}e^{(1+x^2)} + C$
Let $I=\int x e^{(1+x^2)} dx$
Put $t = 1 + x^2$, then $dt = 2x dx ⇒\frac{dt}{2} = x dx$.
$∴I = \int e^t \frac{dt}{2} = \frac{1}{2} \int e^t dt$
$= \frac{1}{2} e^t + C = \frac{1}{2} e^{(1+x^2)} + C$