Integration of Hyperbolic functions

1. Integral involving hyperbolic sine

$$ \int \sinh x \ dx= \cosh x +C $$

2. Integral involving hyperbolic cosine

$$ \int \cosh x \ dx= \sinh x+C $$

3. Integral involving hyperbolic tangent

$$ \int \tanh x \ dx= \ln \cosh x+C $$

4. Integral involving hyperbolic cotangent

$$ \int \coth x \ dx= \ln |\sinh x|+C $$

5. Integral involving hyperbolic secant squared

$$ \int \text{sech}^2 x \ dx= \tanh x+C $$

6. Integral involving hyperbolic cosecant squared

$$ \int \text{cosech}^2 x \ dx= −\coth x+C $$

$$ 7. \ \int \text{sech} \ x \tanh x \ dx= −\text{sech} \ x+C $$

$$ 8. \ \int \text{cosech} \ x \coth x \ dx= −\text{cosech} \ x+C $$

Example:

$$ \text{Evaluate} \int x \cosh (x)^2 \ dx $$

Solution:

$$ \text{Let} \ u=x^2. \text{Then}, du=2x dx $$

$$ \int x \cosh (x)^2 \ dx = \int \frac 12 \cosh u \ du $$

$$ = \frac 12 \sinh u + C $$

$$ = \frac 12 \sinh (x^2) + C $$