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If I=int(sinx+sin^3x)/(cos2x) dx=Pcosx+Q...

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

A

`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)`

Text Solution

Verified by Experts

The correct Answer is:
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)`
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