What is f(x) = int xsqrt(x-2) dx if f(2) = 3 ?

1 Answer
Oct 21, 2017

f(x)=(2x(x-2)^(3/2))/3-(4(x-2)^(5/2))/15+3

Explanation:

f(x)=int\ xsqrt(x-2)\ dx

According to integration by parts,
int\ u\ dv=uv-int\ v\ du.

Set u=x and dv=sqrt(x-2)\ dx=(x-2)^(1/2)\ dx. Then, du=dx and v=(2(x-2)^(3/2))/3.

Thus,
f(x)
=int\ xsqrt(x-2)\ dx
=(2x(x-2)^(3/2))/3-int\ (2(x-2)^(3/2))/3\ dx
=(2x(x-2)^(3/2))/3-(4(x-2)^(5/2))/15+C

Now, since f(2)=3,
(2*2(2-2)^(3/2))/3-(4(2-2)^(5/2))/15+C=3
C=3

Thus, the final answer is f(x)=(2x(x-2)^(3/2))/3-(4(x-2)^(5/2))/15+3