How do you find the integral of $cos(log_e(x)) dx$ ?

Question

1750 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-06-08T18:46:36

First, try integration-by-parts with $u=cos(ln(x))$ , $du=-sin(ln(x))*1/x$ , $dv=dx$ , and $v=x$ to get:

$\int cos(ln(x))\ dx=uv-\int v\ du=x cos(ln(x))+\int sin(ln(x))\ dx$ .

Now use integration-by-parts again on this second integral with $u=sin(ln(x))$ , $du=cos(ln(x))*1/x$ , $dv=dx$ , and $v=x$ to get

$\int cos(ln(x))\ dx=x cos(ln(x))+(xsin(ln(x))-\int cos(ln(x))\ dx)$ .

Adding $\int cos(ln(x))\ dx$ to both sides of this last equation, dividing both sides by 2, and including the $+C$ at the very end gives the answer:

$\int cos(ln(x))\ dx=1/2xcos(ln(x))+1/2xsin(ln(x))+C$ .

How do you find the integral of cos(log_e(x)) dxcos(log_e(x)) dx?