Integrate Cosx/1+sinx dx: No Parts Needed

[HermitOfThebes]

Homework Statement

Integrate Cosx/1+sinx dx from 0 to pi/2. "The question does not assume knowledge of integration by parts."

Homework Equations

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.

 

[SteamKing]

Homework Statement

Integrate Cosx/1+sinx dx from 0 to pi/2. "The question does not assume knowledge of integration by parts."
Could it be found using the quotient rule?
If not, is there any way of proving it without using integration by parts?

Homework Equations

The Attempt at a Solution

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?

 

[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.

 

[SteamKing]

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.

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

 

[MisterX]

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

You are right. HermitOfThebes, ignore my post.

 

[FeDeX_LaTeX]

What's the derivative of sine?

Can you see a neat substitution you could make?

 

[HermitOfThebes]

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?

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

 

[pasmith]

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

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)).

 

[HermitOfThebes]

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)).

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

 

[HermitOfThebes]

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

nvm. I see what you're saying.