Question

What is `int arcsinx/sqrt(1+x^2) dx`?

Answer 1

First of all, one might think "oh yeah, `u` substitution would work!", but `1/sqrt(1-x^2)` is the derivative of `arcsinx`, not `1/sqrt(1+x^2)`, so we can't do that.

First, we would have had to turn the denominator into that form.

`=> int (arcsinx)/sqrt(1 - (-x^2))dx`

For this, we require that `-x^2 -> u^2`. However, if we had intuitively chosen to let `u = -x`, then we have `-x*x = u*-u = -u^2`, not `u^2`.

This indefinite integral gives a non-elementary answer.

It is better to evaluate it numerically, i.e. as a definite integral via calculator or Wolfram Alpha.