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

1 Answer
Sep 22, 2016

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.!