If `int(dx)/(cos^(3)xsqrt(sin2x))=a(tan^(2)x+b)sqrt(tanx)+C`, then

Options

A. `(16)/(5)`
B. `(27)/(5)`
C. `(7)/(10)`
D. `(27)/(10)`

Answer

a

Explanation

Let `I=int(1)/(cos^(3)xsqrt(2 sin2x))dx` Then,
`I=(1)/(2)int(1)/(cos^(7//2)xsin^(1//2)x)dx=(1)/(2)int((1)/(cos^(4)x))/((cos^(7//2)x sin^(1//2)x)/(cos^(4)x))dx`
`rArrI=(1)/(2)int(sec^(4)x)/(tan^(1//2)x)dx=(1)/(2)int(1+t^(2))/(sqrt(t))`, where t= tanx
`rArrI=(1)/(2){(t^(1//2))/(1//2)+(t^(5//2))/(5//2)}+krArrI=sqrt(tanx)+(1)/(5)(tanx)^(5//2)+k`
`thereforeA=(1)/(2), B=(5)/(2)andC=(1)/(5)`
Hence , `A+B+C=(1)/(2)+(5)/(2)+(1)/(5)=(16)/(5)`