What is f(x) = int -cos6x -3tanx+cot(x/3) dx if f(pi)=-2 ?

1 Answer
May 27, 2016

f(x) = int( -cos6x -3tanx+cot(x/3)) dx

=> f(x) = -intcos6xdx -3inttanxdx+intcot(x/3) dx

=> f(x) = -1/6sin6x -3ln|secx|+3ln|sin(x/3)| +c ...color(red)"(1)"

,where c= integration constant

Now imposing the given condition f(pi)=-2

-1/6sin(6xxpi) -3ln|secpi|+3ln|sin(pi/3)| +c=-2

1/6xx0-3xx0+3xxln(sqrt3/2)+c=-2

c=-2-3ln(sqrt3/2)

Inserting the value of c in equation color(red)"(1)" we have

f(x) = -1/6sin6x -3ln|secx|+3ln|sin(x/3)| -2-3ln(sqrt3/2)