Question

section 3.6: additional integration te
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question 1
find $\int x e^{4 x} d x$
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Explanation:

Step1: Apply integration by parts formula

The integration by parts formula is $\int u\mathrm{d}v=uv - \int v\mathrm{d}u$. Let $u = x$, then $\mathrm{d}u=\mathrm{d}x$. And let $\mathrm{d}v=e^{4x}\mathrm{d}x$, then $v=\frac{1}{4}e^{4x}$.

Step2: Substitute into the formula

$\int xe^{4x}\mathrm{d}x=x\cdot\frac{1}{4}e^{4x}-\int\frac{1}{4}e^{4x}\mathrm{d}x$.

Step3: Integrate the remaining part

$\int\frac{1}{4}e^{4x}\mathrm{d}x=\frac{1}{16}e^{4x}$.

Answer:

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