Question

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

• A. $xe^{3x} + \frac{x}{9} + C$
• B. $xe^{3x} - \frac{x}{9} + C$
• C. $\frac{x e^{3x}}{3} + \frac{x}{9} + C$
• D. $\frac{x}{3} - \frac{e^{3x}}{9} + C$

Answer 1

$\frac{xe^{3x}}{3} - \frac{e^{3x}}{9} + C$

Answer 2

To evaluate the integral $\int xe^{3x} dx$, we use the integration by parts formula:

$$\int u dv = uv - \int v du$$

We need to choose $u$ and $dv$. Following the LIATE rule (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential), we choose $u=x$ (Algebraic) and $dv=e^{3x} dx$ (Exponential).

1. Identify $u$ and $dv$: Let $u = x$ Let $dv = e^{3x} dx$

2. Find $du$ and $v$: Differentiate $u$ to find $du$: $du = \frac{d}{dx}(x) dx = dx$

   Integrate $dv$ to find $v$: $v = \int e^{3x} dx = \frac{e^{3x}}{3}$

3. Apply the integration by parts formula: Substitute $u$, $v$, and $du$ into the formula $\int u dv = uv - \int v du$:

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

4. Evaluate the remaining integral: The integral $\int e^{3x} dx$ is $\frac{e^{3x}}{3}$.

5. Substitute back and add the constant of integration:

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

Therefore, the correct answer is $\frac{xe^{3x}}{3} - \frac{e^{3x}}{9} + C$.