Expert: Paul Klarreich - 3/21/2008

Question
I am trying to integrate

(sin^3(x))/sqrt cos(x)

I trying to use integration by parts.
I keep coming up with a messy second integral.

I only want to know if this is the correct
method (by parts) or can you suggest another
method.
If this problem can be solved by parts
I will keep working.

Answer
Questioner:   Mark
Category:  Calculus
Private:  No
 
Subject:  Integration of trig function
...............................
Hi, Mark

Your Question:  I am trying to integrate

(sin^3(x))/sqrt cos(x)

I trying to use integration by parts.
I keep coming up with a messy second integral.

>> Do what I do when I have a mess -- call the wife to clean it up. [Oh -- you're not married yet.]

I only want to know if this is the correct
method (by parts) or can you suggest another
method.

>> I sure can.  See below.

If this problem can be solved by parts
I will keep working.
..................................

Try this:

sin^3(x) = sin x sin^2(x) = sin x(1 - cos^2(x))

{  sin^2 (1 - cos^2)   << getting lazy, I know.
| --------------------- dx
} sqrt(cos)

Now let  cos x = u,  du = - sin x dx

{ - (1 - u^2)
| ----------- du
}  sqrt(u)

{   u^2 - 1
| ----------- du
}    u^1/2


{
|  (u^3/2 - u^-1/2) du
}

and from here on, it should be OK.