What is #F(x) = int 3x^2+e^(2-2x) dx# if #F(0) = 1 #? Calculus Techniques of Integration Evaluating the Constant of Integration 1 Answer Alan N. Jan 12, 2017 #F(x) = x^3 +e^2/2 (1-e^(-2x)) +1# Explanation: #int 3x^2+e^(2-2x) dx = 3intx^2 dx+ e^2 int e^(-2x) dx# #= 3*x^3/3 + e^2 int e^(-2x) dx# #int e^(-2x) dx = -1/2e^(-2x)# #:. F(x) = x^3 - e^2/2 * e^(-2x) +C # Since #F(0) =1# # 0^3 - e^2/2 * e^(-2*0) +C =1# #-e^2/2 *1 +C =1# #C= 1+e^2/2# #F(x) = x^3 - e^2/2 * e^(-2x) +1+e^2/2# #= x^3 +e^2/2 (1-e^(-2x)) +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 1312 views around the world You can reuse this answer Creative Commons License