What is F(x) = int sin(3x)-cos^2(4x) dx if F(pi) = 3 ?

1 Answer
Jun 21, 2016

F(x) = -1/3 cos(3x)- 1/16 sin(8x) - x/2 + 8/3 + pi/2

Explanation:

F(x) = int \ sin(3x)-cos^2(4x) \ dx
using cos 2A = 2 cos^2 A - 1, \qquad cos^2 A = (cos 2 A + 1)/(2)

= int \ sin(3x)-(cos(8x) +1)/2 \ dx

= int \ sin(3x)- 1/2 cos(8x) - 1/2 \ dx

= -1/3 cos(3x)- 1/16 sin(8x) - x/2 + C

F(pi) = 3 \implies 3 = 1/3 - pi/2 + C

C = 8/3 + pi/2

\implies F(x) = -1/3 cos(3x)- 1/16 sin(8x) - x/2 + 8/3 + pi/2