Find $\int e^{x^2} (x^5 + 2x^3) dx$.

$\frac{1}{2} x^4 e^{x^2} + C$

The correct answer is Option (3) → $\frac{1}{2} x^4 e^{x^2} + C$

$\int e^{x^2} (x^5 + 2x^3) dx = \int x e^{x^2} (x^4 + 2x^2) dx$

Let $x^2 = t$, then $2x \, dx = dt$.

$= \frac{1}{2} \int e^t (t^2 + 2t) dt$

$f(t)=t^2$

$∴f'(t)=2t$

Using the property $\int e^t (f(t) + f'(t)) dt = e^t f(t) + c$:

$∴\frac{1}{2} e^t \cdot t^2 + c = \mathbf{\frac{1}{2} x^4 e^{x^2} + c}$