What is f(x) = int xsqrt(x-2) dx if f(1)=-2 ?

1 Answer
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.