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 |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\tan a x$: $14.433$