Primitive of x over Hyperbolic Cosine of a x plus 1

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Theorem

$\ds \int \frac {x \rd x} {\cosh a x + 1} = \frac x a \tanh \frac {a x} 2 - \frac 2 {a^2} \ln \size {\cosh \frac {a x} 2} + C$


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:

\(\ds u\) \(=\) \(\ds x\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds 1\) Derivative of Power


and let:

\(\ds \frac {\d v} {\d x}\) \(=\) \(\ds \frac 1 {\cosh a x + 1}\)
\(\ds \leadsto \ \ \) \(\ds v\) \(=\) \(\ds \frac 1 a \tanh \frac {a x} 2\) Primitive of $\dfrac 1 {\cosh a x + 1}$


Then:

\(\ds \int \frac {x \rd x} {\cosh a x + 1}\) \(=\) \(\ds x \paren {\frac 1 a \tanh \frac {a x} 2} - \int \paren {\frac 1 a \tanh \frac {a x} 2} \rd x + C\) Integration by Parts
\(\ds \) \(=\) \(\ds \frac x a \tanh \frac {a x} 2 - \frac 1 a \int \tanh \frac {a x} 2 \rd x + C\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \frac x a \tanh \frac {a x} 2 - \frac 1 a \paren {\frac 2 a \ln \size {\cosh \frac {a x} 2} } + C\) Primitive of $\tanh a x$
\(\ds \) \(=\) \(\ds \frac x a \tanh \frac {a x} 2 - \frac 2 {a^2} \ln \size {\cosh \frac {a x} 2} + C\) simplifying

$\blacksquare$


Also see


Sources

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