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section 3.6: additional in
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question 6
compute the given integral.
∫xe^{3x}dx=
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basic funcs trig
x \frac{□}{□} x^{□} x_{□}
(□) |□| π
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Explanation:

Step1: Apply integration by parts formula

Integration by parts formula is \(\int u dv=uv-\int v du\). Let \(u = x\), \(dv=e^{3x}dx\). Then \(du = dx\), \(v=\frac{1}{3}e^{3x}\).

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Step2: Integrate the remaining integral

\(\int\frac{1}{3}e^{3x}dx=\frac{1}{9}e^{3x}+C\) (using \(\int e^{ax}dx=\frac{1}{a}e^{ax}+C\) with \(a = 3\)).

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Answer:

\(\frac{e^{3x}}{9}(3x - 1)+C\)