Techniques of integration: Completing the Square and Arcsin: Find

By completing the square, we sometimes can match an integrand with those leading to the inverse trig functions. So after factoring, the quadratic becomes

.

We can add and subtract 1 and rearrange:

Finally, we can move the -1 inside the parentheses:

The quantity in the parentheses is a square, so we have

as the new form we can substitute into the original problem:

.

Now we can try substituting and :

We can use the integral leading to the Arcsin, with a=1:

Now we can use our substitution to get a result in terms of x: