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

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Answer 1 · 2016-06-21T22:43:18

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

#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#