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 1853 views around the world You can reuse this answer Creative Commons License