What is #f(x) = int sin^2x-cotx dx# if #f((5pi)/4) = 0 #? Calculus Techniques of Integration Evaluating the Constant of Integration 1 Answer Ratnaker Mehta Feb 22, 2017 #f(x)=x/2-(sin2x)/4-ln|sinx|+1/4-5pi/8-1/2ln2.# Explanation: #f(x)=int(sin^2x-cotx)dx.# #:. f(x)=intsin^2xdx-intcotxdx,# #=int(1-cos2x)/2dx-ln|sinx|,# #=1/2{int1dx-intcos2xdx}-ln|sinx|,# #rArr f(x)=x/2-(sin2x)/4-ln|sinx|+C.# #"To determine "C," we use the given cond. that "f(5pi/4)=0.# #rArr 5pi/8-(sin(5pi/2))/4-ln|sin(5pi/4)|+C=0.# #rArr 5pi/8-1/4-ln|-1/sqrt2|+C=0.# #:. C=1/4-5pi/8-1/2ln2.# Hence, #f(x)=x/2-(sin2x)/4-ln|sinx|+1/4-5pi/8-1/2ln2.# Enjoy Maths.! 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 1281 views around the world You can reuse this answer Creative Commons License