# Integration problem

1. Sep 12, 2015

### HermitOfThebes

1. The problem statement, all variables and given/known data
Integrate Cosx/1+sinx dx from 0 to pi/2. "The question does not assume knowledge of integration by parts."
2. Relevant equations

3. The attempt at a solution
Could it be found using the quotient rule?
If not, is there any way of proving it without using integration by parts?

Mod note: Edited this post by moving text, to comply with our rules about including an attempt.

Last edited by a moderator: Sep 12, 2015
2. Sep 12, 2015

### SteamKing

Staff Emeritus
The quotient rule is for taking the derivative of the quotient of two functions. There is no quotient rule for integration as such.

Since IBP is off limits, look at the relationship between cos (x) and (1 + sin (x)). Notice anything special?

3. Sep 12, 2015

### MisterX

I assume you mean $\frac{cos(x)}{1+ sin(x)}$.
Immediately I came up with this trick you might try.
$\frac{cos(x)}{1+ sin(x)} \frac{1 - sin(x)}{1 - sin(x)} = \frac{cos(x)(1+sin(x))}{(1-sin^2 x)} = ??$
These integrals can be solved by substitution. Integration by parts is not necessary. There are essentially no general product or quotient rules for integrals besides integration by parts. But this particular integral can be solved with other techniques such as substitution.

4. Sep 12, 2015

### SteamKing

Staff Emeritus
This is much more complicated than just checking out the relationship between the cosine and (1 + sine).

5. Sep 12, 2015

### MisterX

You are right. HermitOfThebes, ignore my post.

6. Sep 12, 2015

### FeDeX_LaTeX

What's the derivative of sine?

Can you see a neat substitution you could make?

7. Sep 13, 2015

### HermitOfThebes

I know that sinx/1+cosx is tan(x/2). I can't quite see the relationship though.

8. Sep 13, 2015

### pasmith

What is $\frac{d}{dx}(1 + \sin x)$?

(Also, please use brackets: sin(x)/1 + cos(x) means $\frac{\sin(x)}{1} + \cos(x)$. You want sin(x)/(1 + cos(x)).

9. Sep 13, 2015

### HermitOfThebes

d/dx (1+sinx) = cosx. But why would I differentiate?

10. Sep 13, 2015

### HermitOfThebes

nvm. I see what you're saying.

Last edited: Sep 13, 2015