TITLE: Integration of $\sqrt { x^2 + a^2 } $ by substitution.
QUESTION [5 upvotes]: When we take $x=a\tan \theta$, then $\sqrt { x^2 + a^2 } =a\mid\sec\theta\mid$. Why do we not use the modulus, why do we simply write $\sec\theta$?

REPLY [6 votes]: Here $\theta$ is (implicitly) between $-\pi/2$ and $\pi/2$. We really are making the ordinary substitution $\theta=\arctan(x/a)$.  On the interval $(-\pi/2,\pi/2)$ the secant is positive.
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