Question

What is `f(x) = int tanx-secx dx` if `f(pi/4)=-1`?

Answer 1

`f(x)=-ln|cosx|-ln|secx+tanx|-ln(sqrt2/(sqrt2+1))-1`

Explanation:

As `inttanxdx=-ln|cosx|` and `intsecxdx=ln|secx+tanx|`

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

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

Hence `f(pi/4)=-ln|1/sqrt2|-ln|sqrt2+1|+c=-1`

or `ln(sqrt2)-ln(sqrt2+1)+c=-1`

or `ln(sqrt2/(sqrt2+1))+c=-1`

or `c=-ln(sqrt2/(sqrt2+1)))-1`

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

= `-ln|cosx|-ln|secx+tanx|-ln(sqrt2/(sqrt2+1))-1`