TITLE: Symmetry during integration
QUESTION [1 upvotes]: What does it mean to 'use the symmetry' in an integration problem? 
I was given the following problem: $$ \int_0^{\frac{\pi}{4}} \log(1+\tan \theta) \ d\theta $$
My friend then advised me to 'use the symmetry'

REPLY [0 votes]: I've always used that for definite integrals we have $\int_a^b f(x)dx=\int_a^b f(b-a-x)dx$.  First let your integral equal I and use mentioned fact, $$I=\int_0^{\frac{\pi}{4}} \log(1+\tan \theta) \ d\theta=\int_0^{\frac{\pi}{4}} \log(1+\tan( \frac{\pi}{4}-\theta)) \ d\theta.$$  Now use the tangent subtraction formula to get: $$\int_0^{\frac{\pi}{4}} \ln(1+\frac{\tan(\frac{\pi}{4})-\tan(\theta)}{1+\tan(\frac{\pi}{4}) \tan(\theta)} )d\theta=\int_0^{\frac{\pi}{4}}\ln(\frac{2}{\tan(\theta)+1})d\theta=I.$$  Adding the first and the last expression and using log rules one has, $$2I=\int_0^{\frac{\pi}{4}}\ln(2)d\theta=\frac{\pi \ln(2)}{4},$$ thus $$I= \frac{\pi \ln(2)}{8}.$$   
Here's where you can find the tangent subtraction formula I used:
http://mathworld.wolfram.com/TrigonometricAdditionFormulas.html
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