Step-by-Step Guide to Evaluating a Tricky Definite Integral: sinx + 2cosx + 3

[twalker40]

1. Evaluate exactly (in terms of $$\pi$$) the definite integral $$\int^{\pi/2}_{-\-\pi/2}$$ $$\frac{dx}{sinx + 2cosx +3}$$

Homework Equations

How do i do this? Step by step instructions if possible.

The Attempt at a Solution

I've tried to manipulate the integral but still don't get anything. I also set the denominator as u. but then i cannot substitute du.

Help??

 

[Roberto Bramb]

Substitute trig. objects with parametric formulae:
sinx=2t/(1+t^2), cosx=(1-t^2)/(1+t^2), dx=2dt/(1+t^2)
and your trig. integrale becomes the rational integral
2*Integ[-1,1](1/(t^2+2*t+5)dt = pi/4

 

[HallsofIvy]

That's clever. Took me a moment to figure out why dx= 2dt/(1+t^2).

 

[Dick]

It's the 'tan(x/2)' substitution in http://en.wikipedia.org/wiki/Trigonometric_substitution

 

[twalker40]

awesome... thanks a lot guys.

We didn't spend that much time on that identity so it totally slipped my mind. Thanks for the reminder!