i ended up with the same thing as you. Here is my reasoning.
u = sin(x) u'=cos(x)
I must change the function in terms of u, so I start with the denominator.
Now I will change my numerator cosx, in terms of u. Using the trigonometric Identity
so now I have ....
Now I must change the constant multiplier of dx. I am going to use
Using & du or u'=cosx, I set them equal to eachother. NOTE: I must also change u' in terms of u
So now I have dx in terms of u, Notice that cosx still has an x. I must remember to keep everything in terms of u. So remembering back to the trigonometric identity, I now have
I plug that back into my new integral
Can I just point out a shorthand approach, using the formula:
? Obviously, working at developing your ability to use the u-substitution is highly beneficial, so I'm not discouraging you from using it, but this approach would undoubtedly save you time and energy. Here, , and
Edit: You could also use the fact that and simply change the sign appropriately.