What is #f(x) = int -2x^2+1/xdx# if #f(2)=-1 #? Calculus Techniques of Integration Evaluating the Constant of Integration 1 Answer Ratnaker Mehta Feb 17, 2017 #f(x)=(-2x^3)/3+ln|x|+13/3-ln2, or,# #f(x)=1/3(13-2x^3)+ln|x/2|.# Explanation: #f(x)=int(-2x^2+1/x)dx=int-2x^2dx+int1/xdx# #=-2intx^2dx+ln|x|=-2(x^(2+1)/(2+1))+ln|x|# #:. f(x)=(-2x^3)/3+ln|x|+C........(star)# But, #f(2)=-1 rArr ((-2)(2)^3)/3+ln|2|+C=-1# #rArr C=16/3-1-ln2=13/3-ln2.# Hence, by #(star)#, #f(x)=(-2x^3)/3+ln|x|+13/3-ln2, or,# #f(x)=1/3(13-2x^3)+ln|x/2|.# 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 1488 views around the world You can reuse this answer Creative Commons License