Question

The integral \( \int xe^x \, dx \) is equal to:

• A. \( xe^x + e^x + C \)
• B. \( e^x - xe^x + C \)
• C. \( x + e^x + C \)
• D. \( xe^x - e^x + C \)
• E. \( xe^x - x^2e^x + C \)

Hint:

For integration by parts, use the formula: \[ \int u \, dv = uv - \int v \, du. \] This method is helpful when the integrand is a product of two functions.

Answer 1

The Correct Option is D

Step 1: We use integration by parts. Let \( u = x \) and \( dv = e^x \, dx \). Then, \( du = dx \) and \( v = e^x \). 
Step 2: Apply the integration by parts formula: \[ \int u \, dv = uv - \int v \, du \] Substituting the values of \( u \) and \( v \): \[ \int xe^x \, dx = x e^x - \int e^x \, dx. \] 
Step 3: Now, integrate \( e^x \): \[ \int e^x \, dx = e^x. \] 
Step 4: Thus, the integral is: \[ \int xe^x \, dx = x e^x - e^x + C. \]