What is #f(x) = int xsqrt(x-2) dx# if #f(1)=-2 #? Calculus Techniques of Integration Evaluating the Constant of Integration 1 Answer Ratnaker Mehta May 10, 2017 # f(x)=2/15(x-2)^(3/2)(3x+4)+C.# Explanation: #f(x)=intxsqrt(x-2)dx.# We subst. #x-2=t^2, i.e., x=2+t^2 rArr dx=2tdt.# #rArr I=int(t^2+2)sqrt(t^2)*2tdt# #=2intt^2(t^2+2)dt=2int(t^4+2t^2)dt# #=2[t^5/5+2*t^3/3]# #=2/15*(3t^5+10t^3)# #=2/15*t^3(3t^2+10)," and, since, "t=sqrt(x-2)," we have,"# # f(x)=2/15*(x-2)^(3/2){3(x-2)+10}+C, or, # # f(x)=2/15(x-2)^(3/2)(3x+4)+C.# 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 1794 views around the world You can reuse this answer Creative Commons License