Primitive of Hyperbolic Secant of a x/Arctangent of Exponential Form

Theorem

$\ds \int \sech a x \rd x = \frac {2 \map \arctan {e^{a x} } } a + C$

Proof

\(\ds \int \sech x \rd x\)

\(=\)

\(\ds 2 \map \arctan {e^x} + C\)

Primitive of $\sech x$: Arctangent of Exponential Form

\(\ds \leadsto \ \ \)

\(\ds \int \sech a x \rd x\)

\(=\)

\(\ds \frac 1 a \paren {2 \map \arctan {e^{a x} } } + C\)

Primitive of Function of Constant Multiple

\(\ds \)

\(=\)

\(\ds \frac {2 \map \arctan {e^{a x} } } a + C\)

simplifying

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sech a x$: $14.626$