# Math Help - Conditional convergence of improper integrals

1. ## Conditional convergence of improper integrals

What is the integral of cos t/t where limits are 1 to x ?

My book says that it is sinx/x - sin1 + integral (from 1 to x) sin t/ t2 dt for all x>=1.

I can't figure out how .

2. ## Re: Conditional convergence of improper integrals

Originally Posted by mrmaaza123
What is the integral of cos t/t where limits are 1 to x ?

My book says that it is sinx/x - sin1 + integral (from 1 to x) sin t/ t2 dt for all x>=1.

I can't figure out how .

Use integration by parts.

$u=\frac{1}{t} \implies du=-\frac{1}{t^2}dt$

and

$dv=\cos(t) \implies v=\sin(t)$

This gives

$\int_{1}^{x} \frac{\cos(t)}{t}dt=\frac{\sin(t)}{t}\bigg|_{t=1}^ {t=x}+\int_{1}^x\frac{\sin(t)}{t^2}dt$

3. ## Re: Conditional convergence of improper integrals

Oh, yes. Thank you, i don't really know how i overlooked that !