Evaluate an Integral - No Clue

[LilTaru]

Evaluate an Integral - No Clue!

Homework Statement

$$\int$$ x/(1 + sin(x)) dx

Homework Equations

The Attempt at a Solution

I don't know what rules to use or what approach to take! Please help!

 

[lanedance]

is this a definite integral? that may simplify it, otherwise i think its pretty complex... can you use a power series approximation?

 

[LilTaru]

No indefinite and yeah looks complex to me...

 

[Dick]

I would start out by multiplying by (1-sin(x))/(1-sin(x)).

 

[lanedance]

nice suggestion Dick

 

[Dick]

nice suggestion Dick

Thanks! I'm not sure how easily it goes from there. Looks like some integration by parts.

 

[LilTaru]

So I got it to x/(1 + sin(x)) = x(1 - sin(x))/(cos²(x)) but now I am stuck again... D:

 

[Lucci]

If you separate (1-sin(x)/cos²(x)) you can do the following...
x(1/cos²(x) - sin(x)/cos²(x)) =
x(sec²(x)-tan(x)sec(x)) =
∫xsec²(x) - ∫xtan(x)sec(x)

Now you can use integration by parts on both integrals?

 

[lurflurf]

hint:recall from elementary trigonomtry that
1+sin(x)=2 sin(x/2+pi/4)^2