Question

Evaluate $\int xe^{3x} dx$ using integration by parts

• A. $xe^{3x} + \frac{2}{3} + C$
• B. $xe^{3x} - \frac{2}{3} + C$
• C. $\frac{x}{3} + \frac{2}{3} + C$
• D. $\frac{x}{3} - \frac{2}{3} + C$

Answer 1

$\frac{xe^{3x}}{3} - \frac{e^{3x}}{9} + C$ (None of the options provided are correct)

Answer 2

To evaluate $\int xe^{3x} dx$ using integration by parts, use the formula $\int u \, dv = uv - \int v \, du$.

Let $u = x$ and $dv = e^{3x} dx$.
Then $du = dx$ and $v = \int e^{3x} dx = \frac{e^{3x}}{3}$.

Substitute these into the formula:

$\int xe^{3x} dx = x \left(\frac{e^{3x}}{3}\right) - \int \left(\frac{e^{3x}}{3}\right) dx$ $= \frac{xe^{3x}}{3} - \frac{1}{3} \int e^{3x} dx$ $= \frac{xe^{3x}}{3} - \frac{1}{3} \left(\frac{e^{3x}}{3}\right) + C$ $= \frac{xe^{3x}}{3} - \frac{e^{3x}}{9} + C$

None of the given options match this result.