How do you integrate this expression?

Question

Do I solve this using Integration by parts? or How else can I solve this expression.

Answer 1

Try substituting $$x^2=t$$

You will get $$xdx=\frac{d t}{2}$$.

Substituting in original eq

$$\int \frac {t e^t}{2} dt$$

Can You proceed from here? Now it's just by parts.

Answer 2

One can also differentiate under the integral sign.

Observe $$\int x^3 e^{x^2} \ dx = \int \frac{\partial}{\partial t} \left(x e^{tx^2} \right) |_{t=1} \ dx= \frac{\partial}{\partial t} \left(\int xe^{tx^2} \ dx \right)|_{t=1}.$$

To evaluate the integral inside the derivative, one can use u substitution, $u=tx^2,$ and $du = 2xt \ dx$ to see the expression simplifies to

$$\frac{\partial}{\partial t} \left(\frac{e^{tx^2}}{2t} \right)|_{t=1}.$$

Finally evaluate this derivative with the quotient rule to get $$\frac{\partial}{\partial t} \left(\frac{e^{tx^2}}{2t} \right)|_{t=1}= \frac{x^2 e^{tx^2}(2t) -2e^{tx^2}}{(2t)^2} |_{t=1}= \frac{2x^2e^{x^2} -2e^{x^2}}{4}$$