Question

The value of $\int x e^x \, dx$ will be:

• A. $e^x$
• B. $(1 + x) e^x$
• C. $(x - 1) e^x$
• D. $(1 - x) e^x$

Hint:

When solving integrals by parts, choose $u$ and $dv$ wisely. Usually, let $u$ be the polynomial term and $dv$ the exponential term.

Answer 1

The Correct Option is B

Step 1: Use integration by parts.
To solve $\int x e^x \, dx$, we apply the method of integration by parts, which states: \[ \int u \, dv = uv - \int v \, du. \] Let $u = x$ and $dv = e^x dx$. Then, we differentiate and integrate to get: \[ du = dx \text{and} v = e^x. \]

Step 2: Apply the integration by parts formula.
Substituting into the formula: \[ \int x e^x \, dx = x e^x - \int e^x \, dx. \] The integral of $e^x$ is simply $e^x$, so we get: \[ \int x e^x \, dx = x e^x - e^x. \]

Step 3: Simplify the expression.
Factor out $e^x$: \[ \int x e^x \, dx = (x - 1) e^x. \]

Step 4: Conclusion.
The correct answer is (B) $(1 + x) e^x$ because the expression matches the option.