What is #F(x) = int 3x^2-e^(2-x) dx# if #F(0) = 1 #? Calculus Techniques of Integration Evaluating the Constant of Integration 1 Answer Ratnaker Mehta May 15, 2017 #F(x)=x^3+e^(2-x)+1-e^2, or, # #F(x)=x^3+e^2(e^-x -1)+1.# Explanation: #F(x)=int3x^2-e^(2-x)dx.# #=3intx^2dx-inte^2*e^-xdx,# #=3(x^3/3)-e^2inte^-xdx# #F(x)=x^3-e^2*(e^-x/-1)+C=x^3+e^(2-x)+C.# But, #F(0)=1 rArr 0^3+e^(2-0)+C=1.# # rArr C=1-e^2.# Therefore, #F(x)=x^3+e^(2-x)+1-e^2, or, # #F(x)=x^3+e^2(e^-x -1)+1.# 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 1272 views around the world You can reuse this answer Creative Commons License