Primitive of Square of Hyperbolic Cosine of a x/Corollary

Theorem

$\ds \int \cosh^2 a x \rd x = \frac {\sinh 2 a x} {4 a} + \frac x 2 + C$

Proof

\(\ds \int \cosh^2 a x \rd x\)

\(=\)

\(\ds \frac {\sinh a x \cosh a x} {2 a} + \frac x 2 + C\)

Primitive of $\cosh^2 a x$

\(\ds \)

\(=\)

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

Double Angle Formula for Hyperbolic Sine

\(\ds \)

\(=\)

\(\ds \dfrac {\sinh 2 a x} {4 a} + \frac x 2 + C\)

simplifying

$\blacksquare$

Also see

• Primitive of $\sinh^2 a x$
• Primitive of $\tanh^2 a x$
• Primitive of $\coth^2 a x$
• Primitive of $\sech^2 a x$
• Primitive of $\csch^2 a x$

Sources

• 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Back endpapers: A Brief Table of Integrals: $116$.