`int (sec^(2)x)/((secx + tanx)^(9//2))dx` equal to (for some arbitrary constant K)

Options

A. `-(1)/((secx+tanx)^(11//2)){(1)/(11)-(1)/(7)(secx+tanx)^(2)}+K`
B. `(1)/((secx+tanx)^(1//11)){(1)/(11)-(1)/(7)(secx+tanx)^(2)}+K`
C. `-(1)/((secx+tanx)^(11//2)){(1)/(11)+(1)/(7)(secx+tanx)^(2)}+K`
D. `(1)/((secx+tanx)^(11//2)){(1)/(11)+(1)/(7)(secx+tanx)^(2)}+K`

Answer

C

Explanation

`I=int(sec^(2)x)/((secx+tanx)^(9//2))dx`
Let ` sec x +tanx=t`
` or sec x -tan x=1//t`
Now, `(secx tanx+sec^(2)x)dx =dt`
`or secx(secx+tanx)dx=dt`
` or secx dx=(dt)/(t),(1)/(2)(t+(1)/(t))=sec x`
` :. I=(1)/(2)int ((t+(1)/(t)))/(t^(9//2))(dt)/(t)`
`=(1)/(2)int(t^(-9//2)+t^(-13//2))dt`
`=(1)/(2)[(t^(-9//2+1))/(-(9)/(2)+1)+(t^(-13//2+1))/(-(13)/(2)+1)]+K`
`=(1)/(2)[(t^(-7//2))/(-(7)/(2))+(t^(-11//2))/(-(11)/(2))]+K`
`= -(1)/(7) t^(-7//2)-(1)/(11)t^(-11//2)+K`
`= -(1)/(7) (1)/(t^(7//2))-(1)/(11)(1)/(t^(11//2))+K`
`= -(1)/(t^(11//2))((1)/(11)+(t^(2))/(7))+K`
`= -(1)/((secx+tan x)^(11//2)){(1)/(11)+(1)/(7)(sec x+tanx)^(2)}+K`