Question

Evaluate the integral: \[ \int x e^x \, dx \] The correct answer is:

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

Hint:

For integrals involving products of polynomials and exponential functions, use integration by parts. This technique can simplify the problem significantly.

Answer 1

The Correct Option is C

We are asked to evaluate the integral: \[ I = \int x e^x \, dx. \] This is a standard integral that can be solved using integration by parts. Recall the formula for integration by parts: \[ \int u \, dv = uv - \int v \, du. \] Let: \[ u = x \quad \text{and} \quad dv = e^x \, dx. \] Then, differentiate and integrate: \[ du = dx \quad \text{and} \quad v = e^x. \] Now, apply the integration by parts formula: \[ I = x e^x - \int e^x \, dx. \] The integral of \( e^x \) is simply \( e^x \), so we get: \[ I = x e^x - e^x + c. \] Factoring out \( e^x \): \[ I = e^x (x - 1) + c. \] Thus, the correct answer is \( (C) \).