The value of the integral `int_(0)^(1)(dx)/(x^(2)+2x cos alpha +1),0ltalpha lt pi` is

Options

A. `sin alpha`
B. `alpha sin alpha`
C. `(alpha)/(sin alpha)`
D. `(alpha)/2 sin alpha`

Answer

C

Explanation

Given integral
`=int_(0)^(1) (dx)/((x+cos alpha)^(2)+(1-cos^(2)alpha))`
`=int_(0)^(1)(dx)/((x+cos alpha)^(2)+sin^(2) alpha)`
`=1/(sin alpha)| "tan"^(-1)(x+cos alpha)/(sin alpha)|_(0)^(1)`
`=1/(sin alpha) ["tan"^(-1)(1+cos alpha)/(sin alpha) -"tan"^(-1) (cos alpha)/(sin alpha)]`
`=1/(sin alpha) [tan^(-1) "cot"(alpha)/2-"tan"^(-1)(cot alpha)]`
`=1/( sin alpha) [tan^(-1)"tan"((pi)/2-(alpha)/2)-tan^(-1)"tan"((pi)/2-alpha)]`
`=1/(sin alpha) [((pi)/2-(alpha)/2)-((pi)/2-alpha)]`
`=(alpha)/(2 sin alpha)`