# need help with an integral

• Mar 1st 2007, 03:15 PM
thedoge
need help with an integral
integral[(sin(y)*cos(y))/(9+(cos(y))^4)]

i know the answer, but i am stuck on how to find it accurately.
• Mar 1st 2007, 03:44 PM
Jhevon
Quote:

Originally Posted by thedoge
integral[(sin(y)*cos(y))/(9+(cos(y))^4)]

i know the answer, but i am stuck on how to find it accurately.

Here's one way to do it.

int{sinycosy/(9 + cos^4(y)}dy
= int{sinycosy/[9 + (cos^2(y))^2]}dy

Now we proceed by substitution:

Let u = cos^2y
=> du = -2cosysiny dy
=> (-1/2)du = cosysiny dy
So our integral becomes

(-1/2)*int{1/(9 + u^2)}du
= (-1/2)*int{1/(3^2 + u^2)}du
= (-1/2)*int{(1/9)*(1/[1 + (u^2)/9])}du
= (-1/2)(1/9)*int{1/(1 + (u/3)^2)}du

Substitution again:

Let v = u/3
=> dv = 1/3 du
=> 3dv = du

So our new integral is:
(-1/18)(3)int{1/(1 + v^2)}dv
= (-1/6)int{1/(1 + v^2)}dv
= (-1/6)arctan(v) + C
= (-1/6)arctan(cos^2(y)/3) + C

This method is correct, but there are other ways to do it that will end up with answers that look different. If you type this function in the "integrator" for instance, the answer looks different from what we have here. But any difference in the answer can be resolved by algebraic manipulation, if not, the difference is absorbed in the constant.
• Mar 1st 2007, 04:19 PM
Jhevon
Quote:

Originally Posted by thedoge
integral[(sin(y)*cos(y))/(9+(cos(y))^4)]

i know the answer, but i am stuck on how to find it accurately.

You got it right? Did you follow all my steps ok? 'cause i dont want you to say thanks if what i did didnt help
• Mar 1st 2007, 04:26 PM
thedoge
yea. i got it right. i was just confused about how to approach the middle section to reach the point where we got the integral to int(1/(1+x^2))=arctan... :) you cleared up for me, so i said thanks.