Question

Evaluate the integral \[ \int \sum_{r=0}^{\infty} \frac{x^r 3^r}{2r} dx. \]

• A. \( e^x + c \)
• B. \( \frac{e^{3x}}{3} + c \)
• C. \( 3e^{3x} + c \)
• D. \( 3e^x + c \)

Hint:

Recognizing standard series expansions can simplify integration.

Answer 1

The Correct Option is B

Step 1: Recognizing the summation Rewriting, \[ \sum_{r=0}^{\infty} \frac{x^r 3^r}{2r} \] resembles the Taylor series of \( e^{3x} \), leading to: \[ I = \int e^{3x} dx. \] Step 2: Evaluating the integral \[ I = \frac{e^{3x}}{3} + c. \]