Question

Evaluate the integral \[ \int e^x (x+1)^2 dx. \]

• A. \( xe^x + c \)
• B. \( e^x x^2 + c \)
• C. \( e^x (x^2 + 1) + c \)
• D. \( e^x (x + 1) + c \)

Hint:

For integrals involving \( e^x \) multiplied by polynomials, use integration by parts recursively.

Answer 1

The Correct Option is C

Step 1: Expanding the function Expanding \( (x+1)^2 \): \[ (x+1)^2 = x^2 + 2x + 1. \] Rewriting the integral: \[ I = \int e^x (x^2 + 2x + 1) dx. \] Step 2: Splitting into separate integrals \[ I = \int e^x x^2 dx + 2\int e^x x dx + \int e^x dx. \] Using integration by parts, where \( u = x^2 \), \( dv = e^x dx \): \[ du = 2x dx, \quad v = e^x. \] Applying integration by parts repeatedly: \[ \int x^2 e^x dx = e^x (x^2 - 2x + 2). \] Similarly, \[ \int x e^x dx = e^x (x - 1). \] Step 3: Evaluating and summing terms \[ I = e^x (x^2 + 1) + c. \]