Evaluate: `int(sin2x)/((1+sinx)(2+sinx)\ dx`

Explanation

`I=int frac{sin 2 x d x}{(1+sin x)(2+sin x)} `

`=int frac{2 sin x cos x d x}{(1+sin x)(2+sin x)}`

Putting `sin x=t`

`Rightarrow cos x d x=d t` `therefore I=int frac{2 t d t}{(1+t)(2+t)}` `=2 int frac{t d t}{(1+t)(2+t)}`
Let `frac{t}{(1+t)(2+t)}=frac{A}{1+t}+frac{B}{2+t}` `Rightarrow frac{t}{(1+t)(2+t)}=frac{A(2+t)+B(1+t)}{(1+t)(2+t)}` `Rightarrow t=A(2+t)+B(1+t)`

`Rightarrow frac{t}{(1+t)(2+t)}=frac{A(2+t)+B(1+t)}{(1+t)(2+t)}`

Putting `2+t=0`

`Rightarrow t=-2 `

`-2=A times 0+B(-2+1) `

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