Primitive of Square of Secant of a x over Tangent of a x

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Theorem

$\ds \int \frac {\sec^2 a x \rd x} {\tan a x} = \frac 1 a \ln \size {\tan a x} + C$


Proof

\(\ds \frac {\d} {\d x} \tan x\) \(=\) \(\ds \sec^2 x\) Derivative of Tangent Function
\(\ds \leadsto \ \ \) \(\ds \int \frac {\sec^2 x \rd x} {\tan x}\) \(=\) \(\ds \ln \size {\tan a x} + C\) Primitive of Function under its Derivative
\(\ds \leadsto \ \ \) \(\ds \int \frac {\sec^2 a x \rd x} {\tan a x}\) \(=\) \(\ds \frac 1 a \ln \size {\tan a x} + C\) Primitive of Function of Constant Multiple

$\blacksquare$


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Sources