Primitive of Cube of Secant of a x/Proof 1
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Theorem
- $\ds \int \sec^3 a x \rd x = \frac 1 {2 a} \paren {\sec a x \tan a x + \ln \size {\sec a x + \tan a x} } + C$
Proof
\(\ds \int \sec^3 x \rd x\) | \(=\) | \(\ds \frac {\sec a x \tan a x} {2 a} + \frac 1 2 \int \sec a x \rd x\) | Primitive of $\sec^n a x$ where $n = 3$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sec a x \tan a x} {2 a} + \frac 1 2 \paren {\frac 1 a \ln \size {\sec a x + \tan a x} }\) | Primitive of $\sec a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a} \paren {\sec a x \tan a x + \ln \size {\sec a x + \tan a x} } + C\) | simplifying |
$\blacksquare$