1. ## Power Series Approximation

Use a power series to approximate
$
\int cos(4x)ln(x) dx.
$

to six decimal places.

Bounds are from pi to 2pi, if you care, but I'm not asking for you to solve the problem completely; just give me the steps.

Thank you so much!

2. $\int\cos 4x\log x dx = \frac{1}{4}\sin 4x\log x - \frac{1}{4}\int \frac{\sin 4x}{x}dx$ and use that $\sin x = \sum_{k=0}^\infty(-1)^k\frac{(4x)^{2k+1}}{(2k+1)!}\Rightarrow \frac{\sin x}{x} = \sum_{k=0}^\infty(-1)^k\frac{4^{2k+1}x^{2k}}{(2k+1)!}$