फलन `f''(x)=sec^(4)x+4` पर, जिसके लिए `f(0)=0` और `f'(0)=0` है विचार कीजिए।
`f(x)` किसके बराबर है ?

Options

A. `(2In sec x)/(3)+(tan^(2)x)/(6)+2x^(2)`
B. `(3In sec x)/(2)+(cot^(2)x)/(6)2x^(2)`
C. `(4In sec x)/(3)+(sec^(2)x)/(6)2x^(2)`
D. `(4In sec x)/(3)+(sec^(2)x)/(6)2x^(2)`

Explanation

`f(x)= intf'(x)dx+C_(2)`
`= int (tan x+(1)/(3)tan^(3)x+4x)dx+C_(2)`
`int tan x dx +(1)/(3) int tan x (sec^(2)x-1)dx+(4x^(2))/(2)+Ca`
`=(2)/(3) int tan x dx +(1)/(3) int tan x* sec^(2) x dx +2x^(2)+C_(2)`
`=(2)/(3) log (secx)+(1)/(3)+I_(2)+ 2x^(2)+C_(2)`
अब `I_(2) int tan x*sec^(2)x dx`
`:. tan x=t` रखने पर `sec^(2)x dx=dt`
`:. I_(2) = int tdt=(t^(2))/(2)+C_(3)`
`rArr I_(2)=(1)/(2) tan^(2) x+C_(3)`
`:. f(x)=(2)/(3)In (sec x)+(1)/(6)tan^(2)x+2x^(2)C_(4)`
जहाँ `C_(4)=C_(2)+(C_(3))/(3)+C_(3)`
तब `0=(2)/(3) log (1)+0+0+C_(4)`
`rArr C_(4)=0`
`:. f(x)=(2)/(3) In (secx)+(1)/(6)tan^(2)x+2x^(2)`