Question

ol2: problem 3
(1 point)
evaluate the indefinite integral.
\\(\int xe^{2x} dx = \square + c\\).
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Explanation:

Step1: Choose integration by parts

Recall integration by parts: $\int u \, dv = uv - \int v \, du$
Let $u = x$, $dv = e^{2x}dx$

Step2: Compute du and v

$du = dx$, $v = \frac{1}{2}e^{2x}$

Step3: Apply integration by parts

$\int xe^{2x}dx = x \cdot \frac{1}{2}e^{2x} - \int \frac{1}{2}e^{2x}dx$

Step4: Evaluate remaining integral

$\int \frac{1}{2}e^{2x}dx = \frac{1}{4}e^{2x}$

Step5: Combine terms

$\int xe^{2x}dx = \frac{1}{2}xe^{2x} - \frac{1}{4}e^{2x}$

Answer:

$\frac{1}{2}xe^{2x} - \frac{1}{4}e^{2x}$