Even More Integration?

Question

I need to integrate

sin(x)
------------------dx
sqrt | cos(x) |

on the interval from pi/2 to 3pi/4.

(The bottom half of that fraction is the square root of the absolute value of cosine of x.)

Should I use substitution or integration by parts? Or is there a better method other than those two?

Thank you!

Answer 1

I = ∫ (sin x / √cos x) dx
let u = cos x (substitution method)
du = - sin x dx
I = - ∫ du / u^(1/2)
I = - ∫ u^(- 1/2) du
I = - 2 u^(1/2) + C
I = - 2 (cos x)^(1/2) + C
I = - 2√(cos x) + C

Answer 2

I would break up the integral into two parts, one where cos x is positive, and the other where it is negative, and then use substitution on each of those integrals. That's what you should always try to do when you have absolute values in the integrand.

Answer 3

I don't have time to go thru it but try:

Int [ cosx ^(-1/2) sin x dx ]

and try it by parts

Answer 4

can you do it like this:
sinx(| cos(x) |)^(-1/2)
use f '(x)[f(x)]^n rule

integral= -2( | cos(x) |)^1/2