How to solve this definite integral?

1. May 1, 2014

PhysicoRaj

1. The problem statement, all variables and given/known data

Integrate:$$I=\int_{-π/4}^{π/4} \ln{(\sec θ-\tan θ)}\,dθ$$

2. Relevant equations

Properties of definite integrals, basic integration formulae, trigonometric identities.

3. The attempt at a solution
By properties of definite integrals, the same integral I wrote as equivalent to$$I=\int_{-π/4}^{π/4} \ln{(\sec θ+\tan θ)}\,dθ$$.
Because$$\int_{a}^{b} f(x)\,dx=\int_{a}^{b} f(a+b-x)\,dx$$(replacing θ by π/4-π/4-θ) Now, I think of adding these two integrals to form an equation and solving for $I$ but I'm messing up. Am I doing wrong? Is there any better/easy way?
Thanks for your time and help.

2. May 1, 2014

Pranav-Arora

What do you get if you add them? It's quite obvious.

Instead, you can notice that the integrand is an odd function, do you see why?

3. May 1, 2014

PhysicoRaj

I tried that at first but I couldn't come to a decision, I can't see if it's odd or even. The secant is an even function right?

4. May 1, 2014

Pranav-Arora

Yes, secant is an even function.

Define $f(\theta)=\ln\left(\sec\theta-\tan\theta\right)$. Hence, $f(-\theta)=\ln\left(\sec\theta+\tan\theta\right)$. What do you get if you add $f(\theta)$ and $f(-\theta)$? Please show the attempt.

5. May 1, 2014

PhysicoRaj

It's $ln(1)=0$

6. May 1, 2014

Pranav-Arora

Right!

So $f(\theta)+f(-\theta)=0$. Do you see why it is an odd function? :)

7. May 1, 2014

PhysicoRaj

Ahh.. I get it right now Pranav, thanks a lot!