Question

section 3.6: additional integra
score: 30/60 answered: 3/6
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question 4
integrate $\int - 4 x e ^ { 9 x } d x$
answer:

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 \(u = - 4x\), \(\mathrm{d}v=e^{9x}\mathrm{d}x\). Then \(\mathrm{d}u=-4\mathrm{d}x\), and \(v=\frac{1}{9}e^{9x}\) (since \(\int e^{ax}\mathrm{d}x=\frac{1}{a}e^{ax}+C\) for \(a = 9\)).

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Step2: Integrate the remaining integral

We know that \(\int e^{9x}\mathrm{d}x=\frac{1}{9}e^{9x}+C\). So,

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Answer:

\(-\frac{4}{9}xe^{9x}+\frac{4}{81}e^{9x}+C\)