Evaluate the integral $\int x e^{2x} dx$ using integration by parts.

Question

Accepted Answer

The formula for integration by parts is:
$$\int u \, dv = uv - \int v \, du$$

**Step 1: Choose $u$ and $dv$ using ILATE rule**
- Let $u = x \implies du = dx$
- Let $dv = e^{2x} dx \implies v = \frac{1}{2} e^{2x}$

**Step 2: Apply the integration by parts formula**
$$\int x e^{2x} dx = (x)\left(\frac{1}{2}e^{2x}\right) - \int \left(\frac{1}{2}e^{2x}\right) dx$$
$$= \frac{1}{2} x e^{2x} - \frac{1}{2} \int e^{2x} dx$$

**Step 3: Complete integration**
$$= \frac{1}{2} x e^{2x} - \frac{1}{2} \left( \frac{1}{2} e^{2x} \right) + C$$
$$= \frac{1}{2} x e^{2x} - \frac{1}{4} e^{2x} + C = \frac{1}{4} e^{2x} (2x - 1) + C$$

Evaluate the integral $\int x e^{2x} dx$ using integration by parts. - DDCET Q&A

Evaluate the integral $\int x e^{2x} dx$ using integration by parts.

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