Question

Evaluate: `int(sinx)/(sqrt(4cos^2x-1))dx`
A. `-(1)/(2)log |2cosx+sqrt(4cos^(2)x-1)|+C`
B. `-(1)/(3)log |2cosx+sqrt(4cos^(2)x-1)|+C`
C. `-(1)/(6)log |cosx+sqrt(2cos^(2)x-1)|+C`
D. None of these

Answer 1

Correct Answer - A
Put ` cos x=t and sin x dx =-dt ,` then
`I=-int(dt)/(sqrt(4t^(2)-1))=-(1)/(2)int (dt)/(sqrt(t^(2)-(1)/(4)))=-(1)/(2)log |t+sqrt(t^(2)-(1)/(4))|+C`
`=-(1)/(2)log |2t+sqrt(4t^(2) -1)|+C=-(1)/(2) log|2cos x+sqrt(4 cos^(2)x-1)|+C.`