Question:
Evaluate the integral
\[
\int \sum_{r=0}^{\infty} \frac{x^r 3^r}{2r} dx.
\]
Evaluate the integral
\[
\int \sum_{r=0}^{\infty} \frac{x^r 3^r}{2r} dx.
\]
Show Hint
Recognizing standard series expansions can simplify integration.
Updated On: Mar 24, 2025
- \( e^x + c \)
- \( \frac{e^{3x}}{3} + c \)
- \( 3e^{3x} + c \)
- \( 3e^x + c \)
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The Correct Option is B
Solution and Explanation
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
\[
I = \frac{e^{3x}}{3} + c.
\]
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