What is F(x) = int sin(3x)-cos^2(4x) dx if F(pi) = 3 ? Calculus Techniques of Integration Evaluating the Constant of Integration 1 Answer Eddie 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 Answer link Related questions How do you find the constant of integration for intf'(x)dx if f(2)=1? What is a line integral? What is f(x) = int x^3-x if f(2)=4 ? What is f(x) = int x^2+x-3 if f(2)=3 ? What is f(x) = int xe^x if f(2)=3 ? What is f(x) = int x - 3 if f(2)=3 ? What is f(x) = int x^2 - 3x if f(2)=1 ? What is f(x) = int 1/x if f(2)=1 ? What is f(x) = int 1/(x+3) if f(2)=1 ? What is f(x) = int 1/(x^2+3) if f(2)=1 ? See all questions in Evaluating the Constant of Integration Impact of this question 1639 views around the world You can reuse this answer Creative Commons License