U-Substitution?

Question

Use U-Substitution to evaluate

5x / (x-2)^1/2 dx

Answer 1

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

Then,

\int 5x/(x-2)^(1/2) dx becomes
\int 5(u+2) * u^(-1/2) du=
5\int (u^(1/2)+2 u^(-1/2)) du=
5 (2/3 u^(3/2)+ 4 u^(1/2)+C=
5 (2/3 (x-2)^(3/2)+ 4 (x-2)^(1/2)+C=

I hope this helps! :)

Answer 2

put

x-2=z^2
then
dx=2zdz

x=z^2+2

the problem now reduces to

(z^2 + 2)/z 2zdz
that is 2*(z^2 + 2)dz
that is (2/3)*z^3 + 2z + C (integrating constant)

now put the values of z

(2/3)*(x-2)^(3/2) + 2*(x-2)^(1/2) + C Ans.

Answer 3

I think that Max P is the one with the right answer, because he took dz for consideration, and organised that data in equations properly.