Primitive of x over Power of Hyperbolic Cosine of a x

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Theorem

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


Proof

\(\ds \int \frac {x \rd x} {\cosh^n a x}\) \(=\) \(\ds \int x \sech^n a x \rd x\) Definition of Hyperbolic Secant
\(\ds \) \(=\) \(\ds \frac 1 {a^2} \int \theta \sech^n \theta \rd \theta\) Substitution of $a x \to \theta$
\(\ds \leadsto \ \ \) \(\ds \frac 1 {a^2} \int \theta \sech^n \theta \rd \theta\) \(=\) \(\ds \frac 1 {a^2} \int \sech^2 \theta \times \theta \sech^{n - 2} \theta \rd \theta\) $\rd u = \sech^2 \theta$ and $v = \theta \sech^{n - 2} \theta$
\(\ds \) \(=\) \(\ds \frac 1 {a^2} \paren {\theta \tanh \theta \sech^{n - 2} \theta - \int \tanh \theta \paren {- \paren {n - 2} \theta \tanh \theta + 1} \sech^{n - 2} \theta \rd \theta}\) Integration by Parts
\(\ds \) \(=\) \(\ds \frac 1 {a^2} \paren {\theta \tanh \theta \sech^{n - 2} \theta - \int \tanh \theta \sech^{n - 2} \theta \rd \theta + \paren {n - 2} \int \theta \tanh^2 \theta \sech^{n - 2} \theta \rd \theta}\) simplifying
\(\ds \) \(=\) \(\ds \frac 1 {a^2} \paren {\theta \tanh \theta \sech^{n - 2} \theta - \int \tanh \theta \sech^{n - 2} \theta \rd \theta + \paren {n - 2} \int \theta \paren {1 - \sech^2 \theta} \sech^{n - 2} \theta \rd \theta}\) Sum of Squares of Hyperbolic Secant and Tangent
\(\ds \) \(=\) \(\ds \frac 1 {a^2} \paren {\theta \tanh \theta \sech^{n - 2} \theta - \int \tanh \theta \sech^{n - 2} \theta \rd \theta + \paren {n - 2} \int \theta \sech^{n - 2} \theta \rd \theta - \paren {n - 2} \int \theta \sech^n \theta \rd \theta}\) simplifying
\(\ds \leadsto \ \ \) \(\ds \frac {1 + \paren {n - 2} } {a^2} \int \theta \sech^n \theta \rd \theta\) \(=\) \(\ds \frac 1 {a^2} \paren {\theta \tanh \theta \sech^{n - 2} \theta - \int \tanh \theta \sech^{n - 2} \theta \rd \theta + \paren {n - 2} \int \theta \sech^{n - 2} \theta \rd \theta}\) adding end term to both sides
\(\ds \leadsto \ \ \) \(\ds \frac 1 {a^2} \int \theta \sech^n \theta \rd \theta\) \(=\) \(\ds \frac 1 {a^2 \paren {n - 1} } \paren {\theta \tanh \theta \sech^{n - 2} \theta - \int \tanh \theta \sech^{n - 2} \theta \rd \theta + \paren {n - 2} \int \theta \sech^{n - 2} \theta \rd \theta}\) rearranging for the intended primitive
\(\ds \) \(=\) \(\ds \frac 1 {a^2 \paren {n - 1} } \paren {\theta \tanh \theta \sech^{n - 2} \theta - \paren {- \frac {\sech^{n - 2} \theta } { \paren {n - 2} } } + \paren {n - 2} \int \theta \sech^{n - 2} \theta \rd \theta}\) Primitive of Power of Hyperbolic Secant of a x by Hyperbolic Tangent of a x
\(\ds \) \(=\) \(\ds \frac {\theta \sinh \theta} {a^2 \paren {n - 1} \cosh^{n - 1} \theta } + \frac 1 {a^2 \paren {n - 1} \paren {n - 2} \cosh^{n - 2} \theta } + \frac {n - 2} {a^2 \paren {n - 1} } \int \frac {\theta} {\cosh^{n - 2} \theta} \rd \theta\) simplifying
\(\ds \leadsto \ \ \) \(\ds \int \frac {x \rd x} {\cosh^n a x}\) \(=\) \(\ds \frac {a x \sinh a x} {a^2 \paren {n - 1} \cosh^{n - 1} a x } + \frac 1 {a^2 \paren {n - 1} \paren {n - 2} \cosh^{n - 2} ax } + \frac {n - 2} {n - 1} \int \frac x {\cosh^{n - 2} ax} \rd x\) Substituting back $\theta \to ax$
\(\ds \) \(=\) \(\ds \frac {x \sinh a x} {a \paren {n - 1} \cosh^{n - 1} a x } + \frac 1 {a^2 \paren {n - 1} \paren {n - 2} \cosh^{n - 2} ax } + \frac {n - 2} {n - 1} \int \frac {x \rd x} {\cosh^{n - 2} ax} + C\) simplifying and adding integration constant

$\blacksquare$


Also see


Sources

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