If `I=int(sinx+sin^3x)/(cos2x)dx=Pcosx+Qlog|f(x)|+R ,` then `P=1/2,Q=-3/(4sqrt(2))` (b) `P=1/4, Q=1/(sqrt(2))` `f(x)=(sqrt(2)cosx+1)/(sqrt(2)cosx-1)` (d) `f(x)=(sqrt(2)cosx-1)/(sqrt(2)cosx+1)`

Options

A. `P=1//2,Q= -(3)/(4sqrt(2))`
B. `P=1//4,Q= -(1)/(sqrt(2))`
C. `f(x)=(sqrt(2)cosx+1)/(sqrt(2)cosx-1)`
D. `f(x)=(sqrt(2)cosx-1)/(sqrt(2)cosx+1)`

Answer

A, C

Explanation

Let `cosx=t " so "cos2x=2t^(2)-1 and dt= -sinx dx.` Thus,
`I=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)/(2sqrt(2))xx(1)/(2)log|(sqrt(2)t-1)/(sqrt(2)t+1)|+C`
`=(1)/(2)cosx-(3)/(4sqrt(2)) log|(sqrt(2)cosx-1)/(sqrt(2)cosx+1)|+C`
So, `P=1//2,Q=-(3)/(4sqrt(2)), f(x)=(sqrt(2)cosx-1)/(sqrt(2)cosx+1)`
`or P=1//2,Q=(3)/(4sqrt(2)),f(x)=(sqrt(2)cosx+1)/(sqrt(2)cosx-1)`