Primitive of Square of Secant of a x

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Theorem

$\displaystyle \int \sec^2 a x \ \mathrm d x = \frac {\tan a x} a + C$


Proof

\(\displaystyle \int \sec^2 x \ \mathrm d x\) \(=\) \(\displaystyle \tan x + C\) Primitive of $\sec^2 x$
\(\displaystyle \implies \ \ \) \(\displaystyle \int \sec^2 a x \ \mathrm d x\) \(=\) \(\displaystyle \frac 1 a \left({\tan a x}\right) + C\) Primitive of Function of Constant Multiple
\(\displaystyle \) \(=\) \(\displaystyle \frac {\tan a x} a + C\) simplifying

$\blacksquare$


Also see


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