# Thread: choosing what method of integration to use?

1. ## choosing what method of integration to use?

$\displaystyle \int sin3x \\\ cosx \\\ dx$

I have been stuck on this question for some time, I initially used integration by parts, but this became very messy and confusing, I had to keep using integration by parts more than twice.

What is the correct method to use? Also I cant really think of any trig manipulation that I can do, to re-write sin3x.

Any help appreciated.

Thank you.

2. Originally Posted by Tweety
$\displaystyle \int sin3x \\\ cosx \\\ dx$

I have been stuck on this question for some time, I initially used integration by parts, but this became very messy and confusing, I had to keep using integration by parts more than twice.

What is the correct method to use? Also I cant really think of any trig manipulation that I can do, to re-write sin3x.

Any help appreciated.

Thank you.
$\displaystyle \sin (a+b) = \sin a \cos b + \sin b \cos a$

$\displaystyle \sin (a-b) = \sin a \cos b - \sin b \cos a$ add them

$\displaystyle \sin (a+b) + \sin (a-b) = 2 \sin a \cos b$

$\displaystyle \sin a \cos b = \left(\frac{1}{2}\right) \left(\sin (a+b) + \sin (a-b) \right)$

use this