Primitive of Reciprocal of Hyperbolic Cosine of a x by 1 plus Hyperbolic Sine of a x
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Theorem
- $\ds \int \frac {\rd x} {\cosh a x \paren {1 + \sinh a x} } = \frac 1 {2 a} \ln \size {\frac {1 + \sinh a x} {\cosh a x} } + \frac 1 a \map \arctan {e^{a x} } + C$
Proof
\(\ds \int \frac {\rd x} {\cosh a x \paren {1 + \sinh a x} }\) | \(=\) | \(\ds \int \frac {\sech^2 a x} {\sech a x + \tanh a x} \rd x\) | Definition of Hyperbolic Cosecant, Definition of Hyperbolic Tangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \int \frac {2 \sech^2 a x} {\sech a x + \tanh a x} \rd x\) | multiplying and dividing by $2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \int \frac {-\sech a x \tanh a x + \sech^2 a x + \sech^2 a x + \sech a x \tanh a x} {\sech a x + \tanh a x} \rd x\) | adding and subtracting $\sech a x \tanh a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \int \frac {-\sech a x \tanh a x + \sech^2 a x} {\sech a x + \tanh a x} \rd x + \frac 1 2 \int \sech ax \paren {\frac {\sech a x + \tanh a x} {\sech a x + \tanh a x} } \rd x\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \int \frac {-\sech a x \tanh a x + \sech^2 a x} {\sech a x + \tanh a x} \rd x + \frac 1 2 \int \sech a x \rd x\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \int \frac {-\sech a x \tanh a x + \sech^2 a x} {\sech a x + \tanh a x} \rd x + \frac 1 a \map \arctan {e^{a x} }\) | Primitive of $\sech a x$: Arctangent of Exponential Form | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a} \int \frac {-a \sech a x \tanh a x + a \sech^2 a x} {\sech a x + \tanh a x} \rd x + \frac 1 a \map \arctan {e^{a x} }\) | multiplying and dividing by $a$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a} \ln \size {\sech a x + \tanh a x} + \frac 1 a \map \arctan {e^{a x} }\) | Derivative of $\sech a x$, Derivative of $\tanh a x$, Primitive of Function under its Derivative | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a} \ln \size {\frac {1 + \sinh a x} {\cosh a x} } + \frac 1 a \map \arctan {e^{a x} } + C\) | simplifying and adding integration constant |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sinh a x$ and $\cosh a x$: $14.601$