If `I=int(sinx+sin^3x)/(cos2x) dx`=`Pcosx+Q log|f(x)|+R`,thenA. `P=(1)/(2), Q=(1)/(4sqrt2)`B. `P=(1)/(4),Q=-(1)/(sqrt2)`C. `f(x)=(cosx+1)/(sqrt2cosx-1)`D. `f(x)=(sqrt2cosx-1)/(sqrt2cosx+1)`

If `I=int(sinx+sin^3x)/(cos2x) dx`=`Pcosx+Q log|f(x)|+R`,thenA. `P=(1)

1.	If `I=int(sinx+sin^3x)/(cos2x) dx`=`Pcosx+Q log\|f(x)\|+R`,thenA. `P=(1)/(2), Q=(1)/(4sqrt2)`B. `P=(1)/(4),Q=-(1)/(sqrt2)`C. `f(x)=(cosx+1)/(sqrt2cosx-1)`D. `f(x)=(sqrt2cosx-1)/(sqrt2cosx+1)`
Answer» Correct Answer - D `l=int(sinx+sin^(3)x)/(cos2x)dx=int(sin x(2-cos^(2)x))/((2cos^(2)x-1))dx` Put `cos x= t rArr dx = -sin x dx` `therefore" "l=int(t^(2)-2)/(2t^(2)-1)dt=(1)/(2)int(2t^(2)-4)/(2t^(2)-1)dt` `=(1)/(2)int dt -(3)/(2)int (dt)/(2t^(2)-1)` `=(1)/(2)t-(3)/(2sqrt2)xx(1)/(2)ln\|(sqrt2t-1)/(sqrt2t+2)\|+C` `=(1)/(2)cos x-(3)/(4sqrt2)ln\|(sqrt 2 cos x-1)/(sqrt2 cos x+1)\|+C` So, `P=1//2, Q=(-3)/(4sqrt2)and f(x)=(sqrt2 cos x-1)/(sqrt2 cos x+1)` or `P=1//2, Q=(3)/(4sqrt2)and f(x)=(sqrt2 cos x+1)/(sqrt2 cos x-1)`