What is f(x) = int -cos^2x dx if f(pi/3) = -6 ?

1 Answer
Sep 1, 2016

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

Explanation:

f(x) = int -cos^2x dx

=- int cos^2x dx

use identity cos 2A = cos^2 A - sin^2 A = 2 cos^2 A - 1

implies - 1/2 int 1 + cos 2 x dx

f(x)= - 1/2 ( x + 1/2sin 2 x) + C

f(pi/3) = -6

implies -6 = - 1/2 ( pi/3 + 1/2sin (2pi )/3) + C

C = -6+sqrt(3)/8+pi/6

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