Find the Taylor series expansion at x = 0 of $\int \frac { x } { 1 + x ^ { 3 } }$ dx. Is $\left[ \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 5 } x ^ { 5 } + \frac { 1 } { 8 } x ^ { 8 } + \frac { 1 } { 11 } x ^ { 11 } \ldots \right] + C$ correct?

Question

Find the Taylor series expansion at x = 0 of

\int \frac { x } { 1 + x ^ { 3 } }

dx. Is

\left[ \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 5 } x ^ { 5 } + \frac { 1 } { 8 } x ^ { 8 } + \frac { 1 } { 11 } x ^ { 11 } \ldots \right] + C

correct?

Quizplus · Accepted Answer

The correct Taylor series expansion at $x = 0$ of $\int \frac{x}{1 + x^3} dx$ is $\left[ \frac{1}{2} x^2 - \frac{1}{5} x^5 + \frac{1}{8} x^8 - \frac{1}{11} x^{11} + \ldots \right] + C$, because the integral of $\frac{x}{1 + x^3}$ can be found by first expanding the integrand as a geometric series $x(1 - x^3 + x^6 - x^9 + \ldots)$ and then integrating term by term.

Find the Taylor Series Expansion at X = 0 Of \int