Mastering Trig Subs: Simplify \int\cos^5(x)dx without the Headache

[silverdiesel]

These trig subs are killing me.

$$\int\cos^5(x)dx$$hints?

 

[George Jones]

$$\cos^{5} x = \left(1 - \sin^2 x \right)^{2} \cos x$$

Regards,
George

 

[silverdiesel]

Thanks George... I am slowly getting the hang of this. I appreciate your help.

 

[HallsofIvy]

One of the first things you should have learned: if you have sine or cosine to an odd power, factor out one of them to use with the dx.

cos⁵ x dx= (cos⁴ x)(cos x dx)
= (cos² x)² (cos x dx)= (1- sin² x)²(cos x dx)

Now, let u= sin x.