Question

Integrate 1/ (1+2(sin x)^2)?

integrate 1/ (1+2(sin x)^2)?

Answer 1

`I=1/sqrt3tan^-1(sqrt3tanx)+c`

Explanation:

Your question after copy pest:
`1/ (1+2(sin x)^2)`

We take,`I=int1/(1+2sin^2x)dx`

`I=int(1/cos^2x)/(1/cos^2x+2sin^2x/cos^2x)dx`

`=intsec^2x/(sec^2x+2tan^2x)dx`

`I=intsec^2x/(1+3tan^2x)dx`

Let, `sqrt3tanx=u=>sqrt3sec^2xdx=du=>sec^2xdx=1/sqrt3du`

`and 3tan^2x=u^2`

`I=int1/(1+u^2)1/sqrt3du`

`=1/sqrt3tan^-1u+c,where,u=sqrt3tanx`

`I=1/sqrt3tan^-1(sqrt3tanx)+c`