## Calculus: The integral of the inverse trigonometric sine function.

Most books in Caculus do not treat the integrals of the

inverse trig and hyperbolic functions, but for completeness,

we include the derivation of the integrals of the inverse

trigonometric functions.

We derive the integral of the inverse sine (sin^{-1}, asin, arcsin) function.

Let . Let us try integration by parts (IBP): .

Performing the substitution, , then .

Also let , then .

Applying IBP, we get,

Consider the second term. Put . Then .

The term then collapses to , which is simply

the derivative of !

The integral of the inverse sine trigonometric sine function is:

.

The derivations of the integrals of the remaining inverse trig functions are done in the same manner.

Let's finish them up soon!

For more derivations of integrals and derivatives, consult Index to Calculus