What is f(x) = int x/sqrt(x-2) dx if f(3)=-1 ?

1 Answer
Mar 22, 2018

f(x) = 2/3(x -2)^(3/2) + 4(x- 2)^(1/2) - 17/3

Explanation:

Let u = x - 2. Then du = dx, and x = u + 2.

f(x) = int x/sqrt(u)dx

f(x) = int (u +2)/sqrt(u) du

f(x) = int u/sqrt(u) + 2/sqrt(u) du

f(x) = int u^(1/2) + 2u^(-1/2) du

f(x) = 2/3u^(3/2) + 4u^(1/2) + C

f(x) = 2/3(x - 2)^(3/2) +4(x- 2)^(1/2) + C

Now we solve for C.

-1 = 2/3(3- 2)^(3/2) + 4(3 -2)^(1/2) + C

-1 = 2/3 + 4 + C

-17/3 = C

Therefore the function is

f(x) = 2/3(x -2)^(3/2) + 4(x- 2)^(1/2) - 17/3

Hopefully this helps!