Integrating Trigonometric Functions: A Helpful Guide

[Redoctober]

Homework Statement

Determine the integral of the following

∫(1/(1+cos(x)).dx
∫(x+sin(x))/(1+cos(x)).dx

The Attempt at a Solution

I tried integration by parts and substution , but didn't work !
Help :/ !

 

[DivisionByZro]

Show us what you've got. Here are hints:
1 - Use the identity that cos(x)=2*cos(x/2)-1, and then use a simple substitution.
2 - Somewhat similar to the first one.

 

[Redoctober]

Show us what you've got. Here are hints:
1 - Use the identity that cos(x)=2*cos(x/2)-1, and then use a simple substitution.
2 - Somewhat similar to the first one.

I got tan(x/2) + C for the first But nothin for the second .
i did as follows

∫(x+sin(x))/(x+cos(x)).dx

simplified it to

∫(x*sec^2(x)).dx + 2∫tan(x/2).dx

I then i integrated them , i used integration by parts for then right hand integral
Then i finally got
2x*tan(x/2)

Is it correct :D ?

 

[SammyS]

Show us what you've got. Here are hints:
1 - Use the identity that cos(x)=2*cos(x/2)-1, and then use a simple substitution.
2 - Somewhat similar to the first one.

That identity should be

cos(x)=2*cos²(x/2)-1 .​

 

[SammyS]

I got tan(x/2) + C for the first But nothin for the second .
i did as follows

∫(x+sin(x))/(x+cos(x)).dx

simplified it to

∫(x*sec^2(x)).dx + 2∫tan(x/2).dx

I then i integrated them , i used integration by parts for then right hand integral
Then i finally got
2x*tan(x/2)

Is it correct :D ?

The ∫(x*sec^2(x)).dx + 2∫tan(x/2).dx that you have should be: ∫(x*sec^2(x/2)).dx + 2∫tan(x/2).dx .

Yes, your answer is correct, if you add an arbitrary constant.

 

[Redoctober]

The ∫(x*sec^2(x)).dx + 2∫tan(x/2).dx that you have should be: ∫(x*sec^2(x/2)).dx + 2∫tan(x/2).dx .

Yes, your answer is correct, if you add an arbitrary constant.

Oh ok thanks :D ! i always forget the arbitrary unit lol xD