Integration using series expansion

[prasoonsaurav]

Can't we integrate tanx/x dx using the series expansion of tan x?

 

[Cyosis]

You can as long as the series converges between the limits of integration.

$$\begin{align*}<br /> \int \frac{\tan x}{x} dx &=\int \frac{1}{x}\sum_{n=1}^\infty \frac{B_{2n} (-4)^n(1-4^n)}{(2n)!} x^{2n-1} dx \\<br /> &=\sum_{n=1}^\infty \left( \frac{B_{2n} (-4)^n(1-4^n)}{(2n)!} \int x^{2n-2} dx \right)<br /> \\<br /> &=\sum_{n=1}^\infty \frac{B_{2n} (-4)^n(1-4^n)}{(2n)!(2n-1)} x^{2n-1} <br /> \\<br /> &=x+\frac{x^3}{9}+\frac{2x^5}{75}+...\;\;\; \text{for} \;|x|<\frac{\pi}{2}<br /> \end{align}$$