The integral `int(sec^2x)/((secx+tanx)^(9/2))dx` equals (for some arbitrary constant `K)dot` `-1/((secx+tanx)^((11)/2)){1/(11)-1/7(secx+tanx)^2}+K` `1/((secx+tanx)^(1/(11))){1/(11)-1/7(secx+tanx)^2}+K` `-1/((secx+tanx)^((11)/2)){1/(11)+1/7(secx+tanx)^2}+K` `1/((secx+tanx)^((11)/2)){1/(11)+1/7(secx+tanx)^2}+K`

Options

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

We have , `I=int(sec^(2)x)/((secx+tanx)^(9//2))dx`
Let `secx-tanx=1//t and, secx(secx+tanx)dx=dt`.
`thereforesecx dx=(1)/(t)dtand, secx=(1)/(2)(t+(1)/(t))`
`thereforeI=(1)/(2)int((1)/(t)(t+(1)/(t)))/(t^(9//2))dt=(1)/(2)int(1)/(t^(9//2))+(1)/(t^(13//2))dt`
`rArr I=-(1)/(7t^(7//2))-(1)/(11t^(11//2))+K`
`rArrI=-(1)/(t^(11//2)){(t^(2))/(7)+(1)/(11)}+K`
`rArr I=-(1)/((secx+tanx)^(11//2)){(1)/(11)+(1)/(7)(secx+tanx)^(2)}+K`