<s>[INST] Evaluation of $\displaystyle \int_{0}^{1}\left(1-x^3+x^5-x^8+x^{10}-x^{13}+\ldots\right)dx$ 
Evaluation of $\displaystyle \int_{0}^{1}\left(1-x^3+x^5-x^8+x^{10}-x^{13}+\ldots\right)dx$

$\bf{My\; try::}$ We can write  $\displaystyle 1-x^3+x^5-x^8+x^{10}-x^{13}+\ldots$ as 
$$\displaystyle (1-x^3)\cdot (1+x^5+x^{10}+\ldots ) = \frac{(1-x^3)(1)}{1-x^5}$$
So we can write it as $\displaystyle \frac{(1-x)(x^2+x+1)}{(1-x)(x^4+x^3+x^2+x+1)}$
So our Integral Convert into $\displaystyle \int_{0}^{1}\frac{x^2+x+1}{x^4+x^3+x^2+x+1}dx$ 
Now How can I solve it, Help me
Thanks 
 [/INST] Hint:
$$
\begin{align}\displaystyle \int_{0}^{1}\left(1-x^3+x^5-x^8+x^{10}-x^{13}+\ldots\right)dx&=
1-{1\over4}+{1\over6}-{1\over9}+{1\over11}-{1\over14}+\ldots\\&=
\sum_{k=0}^\infty{3\over(5k+1)(5k+4)}\end{align}
$$
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