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Integrals of Hyperbolic Functions

Function Integral
sinhx coshx + c
coshx sinhx + c
tanhx ln| coshx | + c
cschx ln| tanh(x/2) | + c
sechx arctan(sinhx) + c = tan-1(sinhx) + c
cothx ln| sinhx | + c

The 6 basic hyperbolic functions are defined by:


Example 1: Evaluate the integral sech2(x)dx

Solution:

We know that the derivative of tanh(x) is sech2(x), so the integral of sech2(x) is just:

tanh(x)+c.

Example 2: Calculate the integral .

Solution : We make the substitution: u = 2 + 3sinh x, du = 3cosh x dx. Then cosh x dx = du/3.

Hence, the integral is



Example 3: Calculate the integral  sinh2x cosh3x dx

Solution:
Applying the formulas and , we get:

Example 4: Evaluate ∫ x sech2 x dx.

Solution:




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