Question

What is `f(x) = int secx-tanx dx` if `f((5pi)/3) = 0`?

Answer 1

`f(x)=ln|2(2+sqrt3)(1+sinx)|`.

Explanation:

`f(x)=int(secx-tanx)dx`

`:. f(x)=intsecxdx-inttanxdx`

`=ln|secx+tanx|-ln|secx|`

`=ln|(secx+tanx)/secx|`

`=ln|secx/secx+tanx/secx|`

`=ln|1+sinx|+c`.

To determine the const.`c", we use the given cond. : "f(5pi/3)=0`

`rArr ln|1+sin(5pi/3)|+c=0`

`"Since, "sin(5pi/3)=sin(2pi-pi/3)=-sin(pi/3)=-sqrt3/2`, so,

`c=-ln|1-sqrt3/2|=-ln|(2-sqrt3)/2|=ln|2/(2-sqrt3)|=ln(2(2+sqrt3))`

`:. f(x)=ln|1+sinx|+ln(2(2+sqrt3))`, or,

`f(x)=ln|2(2+sqrt3)(1+sinx)|`.

Enjoy Maths.!