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

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Answer 1 · 2015-04-09T13:45:16

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

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