What is F(x) = int cos^2x-tan^3x+sinx dx if F(pi/3) = 1 ? Calculus Techniques of Integration Evaluating the Constant of Integration 1 Answer A. S. Adikesavan Nov 3, 2016 F(x)=1/2x+1/4 sin 2x-1/2 tan^2x-ln cos x-cos x+9/2-1/8sqrt 3- ln 2-pi/6 Explanation: Use du=(du)/(dx)dx. Now, F(x)=int((1+cos 2x)/2-tan x (sec^2x-1)+sin x) dx =intd(1/2x)+int d((1/2((sin 2x)/2))-int d((tan^2x)/2) +int d(-ln (cos x))+int d(-cos x) =1/2x+1/4 sin 2x-1/2 tan^2x-ln cos x-cos x + C F(pi/3)=1/6pi+1/4 sin (2/3pi)-tan^2(1/3pi)-ln (cos (1/3pi)-cos (1/3pi) +C =1/6pi+1/8sqrt 3 - 3-ln(1/2)-1/2+C =1 So, C=9/2-1/8sqrt 3- ln 2-pi/6 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 1424 views around the world You can reuse this answer Creative Commons License