Question

How do you find the indefinite integral of `int (7/sqrt(1-u^2))du`?

Answer 1

`int 7/sqrt(1-u^2) du = 7 arcsin u +C`

Explanation:

Substitute:

`u = sint`
`du = costdt`

`int 7/sqrt(1-u^2) du = 7int cost/sqrt(1-sin^2t) dt = 7int cost/sqrt(cos^2t) dt`

As the integrand function is defined for `u in (-1,1)`, we have that `t = arcsinu in (-pi/2,pi/2)`.

In this interval `cost > 0` so `sqrt(cos^2t) = cost`

`int 7/sqrt(1-u^2) du = 7 int cost/cost dt = 7int dt = 7t+C = 7 arcsin u +C`