Integrating Hyperbolic Functions

This lesson covers integration of hyperbolic functions, including standard integrals, integrals producing inverse hyperbolic functions, and hyperbolic substitutions.

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1 Standard Integrals

integral of cosh x dx = sinh x + C

integral of sinh x dx = cosh x + C

integral of sech^2 x dx = tanh x + C

integral of cosech^2 x dx = -coth x + C

integral of sech x tanh x dx = -sech x + C

integral of cosech x coth x dx = -cosech x + C

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2 Worked Example 1

Evaluate the integral from 0 to 1 of cosh x dx.

= [sinh x] from 0 to 1 = sinh 1 = (e - e^(-1))/2 (approximately 1.175)

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3 Integrals Using Double-Angle Identities

integral of sinh^2 x dx = (sinh 2x)/4 - x/2 + C (using sinh^2 x = (cosh 2x - 1)/2)

integral of cosh^2 x dx = (sinh 2x)/4 + x/2 + C (using cosh^2 x = (cosh 2x + 1)/2)

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4 Worked Example 2

Evaluate the integral from 0 to ln 2 of sinh^2 x dx.

Using sinh^2 x = (cosh 2x - 1)/2: