Question

The value of the integral \( \int x e^{-x} \, dx \) is:

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

Hint:

When solving integrals using integration by parts, choose \( u \) and \( dv \) such that the resulting integral is simpler to solve.

Answer 1

The Correct Option is A

Step 1: We apply integration by parts. Let:
- \( u = x \), so \( du = dx \)
- \( dv = e^{-x} dx \), so \( v = -e^{-x} \) 

Step 2: Using the integration by parts formula: \[ \int u \, dv = uv - \int v \, du \] Substitute the values: \[ \int x e^{-x} \, dx = -x e^{-x} - \int -e^{-x} \, dx \] 

Step 3: Simplifying the remaining integral: \[ = -x e^{-x} + e^{-x} \] 

Step 4: Factor the result: \[ = -(x + 1) e^{-x} \] Thus, the correct answer is (A) \( -(x + 1) e^{-x} \).