1. ## Integration problem

Hi everyone. I'm having trouble with an integration problem.
The problem is:
Solve the integral for:

sin(4x) dx
_________
4 + (cos(4x))^2

I put u=cos4x and du=-4sin(4x). However, whenever I continue to solve, I just don't come up with the right answer. Can someone work through this problem for me?

2. Originally Posted by etha
Hi everyone. I'm having trouble with an integration problem.
The problem is:
Solve the integral for:

sin(4x) dx
_________
4 + (cos(4x))^2

I put u=cos4x and du=-4sin(4x) dx. However, whenever I continue to solve, I just don't come up with the right answer. Can someone work through this problem for me?
First, the bit in red I added is important - it's omission by you is probably why you're going wrong. Then:

Substitute $dx = - \frac{du}{4 \sin(4x)}$.

I couldn't see the above latex equation, there might be a problem so I'll give it here too: dx = - du/(4 sin(4x))

3. no, I've done this. My answer is still wrong.

4. Show us what you've done and we'll see if we can try to help. Better than us having to go through the whole thing and finding out that you only need one small step to finish.

5. Originally Posted by etha
Hi everyone. I'm having trouble with an integration problem.
The problem is:
Solve the integral for:

sin(4x) dx
_________
4 + (cos(4x))^2

I put u=cos4x and du=-4sin(4x). However, whenever I continue to solve, I just don't come up with the right answer. Can someone work through this problem for me?
Your choice for u is correct: $u=cos(4x)$. Thus $du=-4sin(4x)dx$. Solving for $dx$, we get $dx=\frac{du}{-4sin(4x)}$. Now make your substitutions:

$\int\frac{sin(4x) du}{-4sin(4x)(4+u^2)}$.

The integral now becomes

$-\frac{1}{4}\int\frac{du}{4+u^2}$.

However, recall that

$\int\frac{du}{a^2+u^2}=\frac{1}{a}arctan(\frac{u}{ a})+c$

Therefore,

$-\frac{1}{4}\int\frac{du}{4+u^2}=-\frac{1}{8}\arctan(\frac{u}{2})+c$

Substituting u back into the evaluated integral, we get

$-\frac{1}{8}\arctan(\frac{cos(4x)}{2})+c$.

Hope this helps you out!

6. what I've done so far is u= cos(4x) and du=-4sin(4x) dx. I know that I need to put it in 1/1+u^2 so that i can integrate it as a arctan(x) + c. there's a 1/4 in the original problem I have to account for. At this point, I'm at a loss as to what to do.

7. I understand now. Thank you.