Question

Evaluate: `int(cosx)/(sqrt(sin^2x-2sinx-3))dx`
A. `log |sinx+sqrt(sin^(2)x-2sinx-3)|+C`
B. `log |(sinx-1)+sqrt(sin^(2)x-2sinx-3)|+C`
C. `log |(sinx-1)-sqrt(sin^(2)x-2sinx-3)|+C`
D. None of these

Answer 1

Correct Answer - B
Putting `sin x=t and cos x dx =dt ,` we get
`I=int (dt)/(sqrt(t^(2)-2t-3))=int (dt)/(sqrt((t^(2)-2t+1)-4))=int (dt)/(sqrt((t-1)^(2)-2^(2)))`
`=log |(t-1)+sqrt((t-1)^(2)-2^(2))|+C=log |(t-1)+sqrt(t^(2)-2t-3)|+C`
`=log |(sin x-1)+sqrt(sin^(2)x-2sin x-3)|+C.`