Primitive of Reciprocal of Hyperbolic Cosine of a x plus 1

Theorem

$\ds \int \frac {\d x} {\cosh a x + 1} = \frac 1 a \tanh \frac {a x} 2 + C$

Proof

\(\ds u\)

\(=\)

\(\ds \tanh \frac x 2\)

\(\ds \leadsto \ \ \)

\(\ds \int \frac {\d x} {1 + \cosh x}\)

\(=\)

\(\ds \int \frac {\dfrac {2 \rd u} {1 - u^2} } {\dfrac {1 + u^2} {1 - u^2} + 1}\)

Hyperbolic Tangent Half-Angle Substitution

\(\ds \)

\(=\)

\(\ds \int \frac {2 \rd u} {1 + u^2 + \paren {1 - u^2} }\)

multiplying top and bottom by $1 - u^2$

\(\ds \)

\(=\)

\(\ds \int \rd u\)

simplifying

\(\ds \)

\(=\)

\(\ds u + C\)

Primitive of Constant

\(\ds \)

\(=\)

\(\ds \tanh \frac x 2 + C\)

substituting for $u$

\(\ds \leadsto \ \ \)

\(\ds \int \frac {\d x} {1 + \cosh a x}\)

\(=\)

\(\ds \frac 1 a \tanh \frac {a x} 2 + C\)

Primitive of Function of Constant Multiple

$\blacksquare$

Also see

• Primitive of $\dfrac 1 {\cosh a x - 1}$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\cosh a x$: $14.575$