Subject: Calculus

List Of Integrals Of Area Functions

In Calculus, area functions are also known as Inverse Hyperbolic Functions. The term "area function" got its name because it represents the area of a sector of the unit hyperbola x^2 − y^2 = 1. The following are the area functions;

  1. arsinh (x) = sinh^{-1}(x)
  2. arcosh (x) = cosh^{-1}(x)
  3. artanh (x) = tanh^{-1}(x)
  4. arcoth (x) = coth^{-1}(x)
  5. arcsch (x) = csch^{-1}(x)
  6. arsech (x) = sech^{-1}(x)

Integrals of Area Functions

Integrals of Inverse hyperbolic functions or area functions can be done through integration by parts. But instead of going through the lengthy process of integration by parts, let me just provide you directly with the list of integrals of area functions which are as follows:

1. \int arsinh (x) = \int sinh^{-1}(x)

=x\sinh^{-1}(x)-\sqrt{x^2+1} + C

2. \int arcosh (x) = \int cosh^{-1}(x)

=x\cosh^{-1}(x)-\sqrt{x-1}\sqrt{x+1} + C

3. \int artanh (x) = \int tanh^{-1}(x)

=\frac{1}{2}\log{\left ( 1-x^2\right )} + x \tanh (x)+ C

4. \int arcoth (x) = \int coth^{-1}(x)

=\frac{1}{2}\log{\left ( 1-x^2\right )} + x \coth (x)+ C

5. \int arcsch (x) = \int csch^{-1}(x)

=x\left [ \frac{\sqrt{\frac{1}{x^2}+1}sinh^{-1}(x)}{\sqrt{x^2+1}}+ csch^{-1}(x) \right ] + C

5. \int arsech (x) = \int sech^{-1}(x)

=x sech^{-1}(x) - \frac{2\sqrt{\frac{1-x}{x+1}}\sqrt{1-x^2}\sin^{-1}{\left ( \frac{\sqrt{x+1}}{\sqrt{2}}\right )}}{x-1} + C

< >