Question

How can slove this ? `int sin(2x)/ "1+cot^2(x)" dx`

how can slove this ?
`int sin(2x)/ "1+cot^2(x)" dx`

Answer 1

`I=1/2sin^4(x)+C`

Explanation:

We want to solve

`I=intsin(2x)/(1+cot^2(x))dx`

Use the trig identities

`color(blue)(sin(2x)=2cos(x)sin(x)` and `color(blue)(csc^2(x)=1+cot^2(x)`

Thus

`I=int(2cos(x)sin(x))/(csc^2(x))dx`

`color(white)(I)=2intcos(x)sin^3(x)dx`

Make a substitution `u=sin(x)=>du=cos(x)dx`

`I=2intu^3du`

`color(white)(I)=1/2u^4+C`

Substitute back `u=sin(x)`

`I=1/2sin^4(x)+C`