Primitive of Power of Hyperbolic Secant of a x

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Theorem

$\ds \int \sech^n a x \rd x = \frac {\sech^{n - 2} a x \tanh a x} {a \paren {n - 1} } + \frac {n - 2} {n - 1} \int \sech^{n - 2} a x \rd x + C$

for $n \ne -1$.


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:

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


and let:

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


Then:

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

$\blacksquare$


Also see


Sources

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