Find the Taylor series about $x = 0$ for the indefinite integral $\int \frac { x } { 1 - 5 x ^ { 2 } } d x$
A) $\sum _ { n = 0 } ^ { \infty } \frac { 5 ^ { 2 n } } { 2 n } x ^ { 2 n }$
B) $\sum _ { n = 0 } ^ { \infty } \frac { 5 ^ { n } } { 2 n + 2 } x ^ { 2 n + 2 }$
C) $\sum _ { n = 0 } ^ { \infty } \frac { 5 ^ { n } } { 2 n } x ^ { 2 n }$
D) $\sum _ { n = 0 } ^ { \infty } \frac { 5 ^ { n } } { 2 n + 1 } x ^ { 2 n + 1 }$

Question

Quizplus · Accepted Answer

The integral $\int \frac{x}{1-5x^2} dx$ can be solved by recognizing it as a geometric series expansion. The integrand $\frac{x}{1-5x^2}$ can be rewritten as $x \sum_{n=0}^{\infty} (5x^2)^n$, which upon integration term by term gives $\sum_{n=0}^{\infty} \frac{5^n}{2n+2}x^{2n+2}$, matching choice B.

Find the Taylor Series About x=0x = 0x=0 For the Indefinite Integral

Find the Taylor Series About $x = 0$ For the Indefinite Integral