The value of the integral `int_(0)^(1)(dx)/(x^(2)+2x cos alpha +1),0ltalpha lt pi` isA. `sin alpha`B. `alpha sin alpha`C. `(alpha)/(sin alpha)`D. `(alpha)/2 sin alpha`

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

1.	The value of the integral `int_(0)^(1)(dx)/(x^(2)+2x cos alpha +1),0ltalpha lt pi` isA. `sin alpha`B. `alpha sin alpha`C. `(alpha)/(sin alpha)`D. `(alpha)/2 sin alpha`
Answer» Correct Answer - C 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)`

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The area bounded by the curves `y=cosx` and `y=sinx` between the ordinates `x=0` and `x=(3pi)/2` isA. `4sqrt(2) + 2`B. `4sqrt(2) +1`C. `4sqrt(2) + 1`D. `4sqrt(2) - 2`
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Given that `lim_(nrarroo) sum_(r=1)^(n) (log_(e)(n^(2)+r^(2))-2log_(e)n)/n = log_(e)2+pi/2-2`, then evaluate : ` lim_(nrarroo) (1)/(n^(2m))[(n^(2)+1^(2))^(m)(n^(2)+2^(2))^(m) "......"(2n^(2))^(m)]^(1//n)`.
`lim_(nrarroo) sum_(r=0)^(n-1) (1)/(sqrt(n^(2)-r^(2)))`
`lim_(nrarroo) (((n+1)(n+2)"........"3n)/(n^(2n)))^(1//n)` is equal to :A. `(27)/(e^(2))`B. `(9)/(e^(2))`C. `3 log3-2`D. `(18)/(e^(4))`