# Thread: Indefinite and Definite integrals as Power series

1. ## Indefinite and Definite integrals as Power series

F(x) = integral of [x / (1-x^6)]

Integral from 0 to .3 of [Ln (1+x^3)] dx

I'm so lousy at this stuff. I don't know how to even start these.

And thanks to everyone that helped me before. I really appreciate it. I'm not just looking for answers. I'm really trying to do these... I just CAN'T get it right.

2. Originally Posted by thegame189
F(x) = integral of [x / (1-x^6)]

Integral from 0 to .3 of [Ln (1+x^3)] dx

I'm so lousy at this stuff. I don't know how to even start these.

And thanks to everyone that helped me before. I really appreciate it. I'm not just looking for answers. I'm really trying to do these... I just CAN'T get it right.
here is the first one

let $g(x)=\frac{1}{1-x}=\sum_{n=0}^{\infty}x^n$
then
$g(x^6)=\frac{1}{1-x^6}=\sum_{n=0}^{\infty}x^{6n}$

finally

$F(x)=\frac{x}{1-x^6}=xg(x^6)=x\sum_{n=0}^{\infty}x^{6n}=\sum_{n=0} ^{\infty}x^{6n+1}$

integraing both ends gives

$\int F(x)dx = \int \sum_{n=0}^{\infty} x^{6n+1} dx = C+ \sum_{n=0}^{\infty} \frac{x^{6n+2}}{6n+2}$

3. Originally Posted by thegame189
F(x) = integral of [x / (1-x^6)]

Integral from 0 to .3 of [Ln (1+x^3)] dx

I'm so lousy at this stuff. I don't know how to even start these.

And thanks to everyone that helped me before. I really appreciate it. I'm not just looking for answers. I'm really trying to do these... I just CAN'T get it right.
Hint let $g(x)=\ln(1+x)$

then $g'(x)=\frac{1}{1+x}$

use the same proceedure as above.

Good luck.