Question

How do you integrate `(sin(2x))/(5-sin(x))^(1/2) dx`?

Answer 1

First rewrite : `2int (cos(x)sin(x))/sqrt((5-sin(x))) dx`

Then `u = sin(x)` so `du = cos(x) dx`

Now you have `2int u/sqrt(5-u) du`

now substitute again `t = 5-u` so `dt = -du`

and `u = 5-t`

Now it's more easy : `-2int (5-t)/sqrt(t) dt`

`= -2int 5/sqrt(t)-sqrt(t) dt`

`= 2intsqrt(t) dt - 10int1/sqrt(t) dt`

`= 4/3*t^(3/2) - 20sqrt(t) + C`

Substitute back for `t = 5 - u`

`=4/3(5-u)^(3/2)-20sqrt(5-u) + C`

Again for `u = sin(x)`

`=4/3(5-sin(x))^(3/2)-20sqrt(5-sin(x)) + C`

You can factorize if you want