Primitive of Hyperbolic Secant Function/Arctangent of Exponential Form

Theorem

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

Proof

Let:

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

\(=\)

\(\ds \int \frac 2 {e^x + e^{-x} } \rd x\)

Definition 1 of Hyperbolic Secant

\(\ds \)

\(=\)

\(\ds \int \frac {2 e^x} {e^{2 x} + 1} \rd x\)

multiplying top and bottom by $e^x$

Let:

\(\ds u\)

\(=\)

\(\ds e^x\)

\(\ds \leadsto \ \ \)

\(\ds u'\)

\(=\)

\(\ds e^x\)

Derivative of Exponential Function

Then:

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

\(=\)

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

Integration by Substitution

\(\ds \)

\(=\)

\(\ds 2 \arctan u + C\)

Primitive of $\dfrac 1 {x^2 + a^2}$: Arctangent Form

\(\ds \)

\(=\)

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

Definition of $u$

$\blacksquare$

Also see

• Primitive of Hyperbolic Secant Function/Arcsine Form

• Primitive of Hyperbolic Cosecant Function

Sources

• 1960: Margaret M. Gow: A Course in Pure Mathematics ... (previous) ... (next): Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xxvii)}$
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: General Rules of Integration: $14.29$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Appendix: Table $2$: Integrals
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Appendix: Table $2$: Integrals