Question

evaluate the integral using integration by parts.
int xe^{4x} dx
int xe^{4x} dx=square

Explanation:

Step1: Choose u and dv

Let $u = x$, $dv=e^{4x}dx$.

Step2: Find du and v

Differentiate $u$: $du = dx$. Integrate $dv$: $v=\frac{1}{4}e^{4x}$.

Step3: Apply integration - by - parts formula

The integration - by - parts formula is $\int u\;dv=uv-\int v\;du$. Substitute $u$, $v$, $du$ and $dv$ into the formula:
$\int xe^{4x}dx=x\cdot\frac{1}{4}e^{4x}-\int\frac{1}{4}e^{4x}dx$.

Step4: Integrate the remaining integral

$\int\frac{1}{4}e^{4x}dx=\frac{1}{16}e^{4x}+C$.
So $\int xe^{4x}dx=\frac{1}{4}xe^{4x}-\frac{1}{16}e^{4x}+C$.

Answer:

$\frac{1}{4}xe^{4x}-\frac{1}{16}e^{4x}+C$