Question

`intcosx/((1 + sinx)(2 + sinx))`dx = ______ 

Options

• log|(1 + sin x) (2 + sin x)| + c

• `log|(2 + sinx)/(1 + sinx)| + c`

• `log|(1 + sinx)/(2 + sinx)| + c`

• `log|(2 + cosx)/(1 + sinx)| + c`

Answer 1

`intcosx/((1 + sinx)(2 + sinx))`dx = `underline(log|(1 + sinx)/(2 + sinx)| + c)`

Explanation:

Put sinx = t

⇒ cosx dx = dt

∴ `intcosx/((1 + sinx)(2 + sinx))dx = intdt/((t + 1)(t + 2))`

= `int1/(t + 1)dt - int1/(t + 2)dt`

= log|t + 1| - log|t + 2| + c

= `log|(t + 1)/(t + 2)| + c`

= `log|(sinx + 1)/(sinx + 2)| + c`