What is #f(x) = int sinx-cos^2x dx# if #f(pi/6)=2 #? Calculus Techniques of Integration Evaluating the Constant of Integration 1 Answer maganbhai P. Mar 10, 2018 #f(x)=-[cosx+x/2+1/4*sin2x]+(48+15sqrt(3)+2pi)/24# Explanation: #f(x)=int(sinx-cos^2x)dx# #f(x)=int(sinx-1/2(1+cos2x))dx=[-cosx-1/2(x+(sin2x)/2)]+c# #f(x)=[-cosx-x/2-1/4*sin2x]+c#, #to (1)# #f(pi/6)=[-cos(pi/6)-pi/12-1/4*sin(pi/3)]+c=2=>[-sqrt(3)/2-pi/12-1/4sqrt(3)/2]+c=2=>-[sqrt(3)/2+pi/12+1/4sqrt(3)/2]+c=2##=>-(12sqrt(3)+2pi+3sqrt(3))/24+c=2=>c=2+(15sqrt(3)+2pi)/24# #.c=(48+15sqrt(3)+2pi)/24#, #to (2)# From (1) and (2) #f(x)=-[cosx+x/2+1/4*sin2x]+(48+15sqrt(3)+2pi)/24# 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 1512 views around the world You can reuse this answer Creative Commons License