`Ifint(dx)/(cos^3xsqrt(sin2x))=a(tan^2x+b)sqrt(tanx)+c ,t h e n` `a=(sqrt(2))/5,b=1/(sqrt(5))` `a=(sqrt(2))/5,b=5` `a=(sqrt(2))/5,b=-1/(sqrt(5))` (d) `a=(sqrt(2))/5, b=sqrt(5)`

Options

A. `a=(sqrt(2))/(5), b=(1)/(sqrt(5))`
B. `a=(sqrt(2))/(5), b=5`
C. `a=(sqrt(2))/(5), b= -(1)/(sqrt(5))`
D. `a=(sqrt(2))/(5), b= sqrt(5)`

Answer

B

Explanation

`I=int(dx)/(cos^(3)xsqrt(sin2x))`
`=int(dx)/(cos^(3)xsqrt((2sinx cosx)/(cos^(2)x)cos^(2)x))`
`=int(sec^(4)x)/(sqrt(2tanx))=(1)/(sqrt(2))int(sec^(2)x(1+tan^(2)x))/(sqrt(tanx))dx`
Let `t=sqrt(tanx)`
or `dt=(sec^(2)xdx)/(2sqrt(tanx))`
` :. dt =(2)/(sqrt(2))int(1+t^(4))dt=sqrt(2)(1+(t^(5))/(5))+C`
`=(sqrt(2))/(5) t(t^(4)+5)+C`
`=(sqrt(2))/(5)sqrt(tanx)(tan^(2)x+5)+C`
or `a=(sqrt(2))/(5), b=5`