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**arbolis** The problem states to use the substitution $\displaystyle t=\tan \left( \frac{x}{2} \right)$ or equivalently $\displaystyle x=2 \arctan (t)$ with the integral $\displaystyle \int \frac{dx}{1+\cos (x)}$. I know that the derivative of the $\displaystyle \arctan$ function is $\displaystyle \frac{1}{1+x^2}$ which is very similar to the integral I must calculate, but I don't understand why we bother with a coefficient of $\displaystyle 2$ here... Also, even if it's quite similar, I don't know how to do it. Can you help me a bit? Maybe there's something to do with $\displaystyle \sqrt{\cos (x)}$...