Integrating 1/(1+cosx) from -pi/2 to pi/2

[Math9999]

Homework Statement

Find the integral of 1/(1+cosx) dx from -pi/2 to pi/2.

Homework Equations

None.

The Attempt at a Solution

Here's my work:
1/(1+cosx)=(1-cosx)/((1+cosx)(1-cosx))=(1-cosx)/(1-cos^2 x)=(1-cosx)/sin^2 x
This is what I've got so far. But this doesn't seem to simplify the integrand.

 

[PeroK]

Homework Statement

Find the integral of 1/(1+cosx) dx from -pi/2 to pi/2.

Homework Equations

None.

The Attempt at a Solution

Here's my work:
1/(1+cosx)=(1-cosx)/((1+cosx)(1-cosx))=(1-cosx)/(1-cos^2 x)=(1-cosx)/sin^2 x
This is what I've got so far. But this doesn't seem to simplify the integrand.

That looks good so far. What about splitting up the integral into two now?

 

[Math9999]

Never mind. I got it. You gave me the big hint already, splitting it up.

 

[Math9999]

Thank you so much! Merry Christmas!

 

[PeroK]

Never mind. I got it. You gave me the big hint already, splitting it up.

There was a quicker way. Hint: $x = 2(x/2)$.

 

[Math9999]

Wow.