Primitive of Power of Secant of a x

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Theorem

$\ds \int \sec^n a x \rd x = \frac {\sec^{n - 2} a x \tan a x} {a \paren {n - 1} } + \frac {n - 2} {n - 1} \int \sec^{n - 2} a x \rd x$

where $n \ne 1$.


Proof

With a view to expressing the primitive in the form:

$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u}{\d x} \rd x$

let:

\(\ds u\) \(=\) \(\ds \sec^{n - 2} a x\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds a \paren {n - 2} \sec^{n - 3} a x \sec a x \tan a x\) Derivative of Power, Derivative of $\sec$, Chain Rule for Derivatives
\(\ds \) \(=\) \(\ds a \paren {n - 2} \sec^{n - 2} a x \tan a x\)


and let:

\(\ds \frac {\d v} {\d x}\) \(=\) \(\ds \sec^2 a x\)
\(\ds \leadsto \ \ \) \(\ds v\) \(=\) \(\ds \frac {\tan a x} a\) Primitive of $\sec^2 a x$


Then:

\(\ds \int \sec^n a x \rd x\) \(=\) \(\ds \int \sec^{n - 2} a x \sec^2 a x \rd x\)
\(\ds \) \(=\) \(\ds \sec^{n - 2} a x \paren {\frac {\tan a x} a}\) Integration by Parts
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \int \paren {\frac {\tan a x} a} \paren {a \paren {n - 2} \sec^{n - 2} a x \tan a x} \rd x\)
\(\ds \) \(=\) \(\ds \frac {\sec^{n - 2} a x \tan a x} a - \paren {n - 2} \int \tan^2 a x \sec^{n - 2} a x \rd x\) simplifying
\(\ds \) \(=\) \(\ds \frac {\sec^{n - 2} a x \tan a x} a - \paren {n - 2} \int \paren {\sec^2 a x - 1} \sec^{n - 2} a x \rd x\) Difference of $\sec^2$ and $\tan^2$
\(\ds \) \(=\) \(\ds \frac {\sec^{n - 2} a x \tan a x} a - \paren {n - 2} \int \sec^n a x \rd x\) Linear Combination of Primitives
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {n - 2} \int \sec^{n - 2} a x \rd x\)
\(\ds \leadsto \ \ \) \(\ds \paren {n - 1} \int \sec^n a x \rd x\) \(=\) \(\ds \frac {\sec^{n - 2} a x \tan a x} a + \paren {n - 2} \int \sec^{n - 2} a x \rd x\) gathering terms
\(\ds \int \sec^n a x \rd x\) \(=\) \(\ds \frac {\sec^{n - 2} a x \tan a x} {a \paren {n - 1} } + \frac {n - 2} {n - 1} \int \sec^{n - 2} a x \rd x\) dividing by $n - 1$

$\blacksquare$


Also see


Sources

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