What is f(x) = int -cos^2x dx if f(pi/3) = 0 ?

1 Answer
Jan 31, 2017

f(x)=pi/6+sqrt(3)/8-1/2x-1/4sin2x

Explanation:

int-cos^2x dx
=-1/2int(1+cos2x)dx
=-1/2x-1/4sin2x+c
But
f(pi/3)=0
Therefore
0=-1/2(pi/3)-1/4sin(2pi/3)+c
0=-pi/6-1/8sqrt(3)+c
So
c=pi/6+sqrt(3)/8

Notes:
cos2x=cos^2x-sin^2x
=cos^2x-(1-cos^2x)=2cos^2x-1
sin((2pi)/3)=sqrt(3)/2