Question

Can someone find the indefinate integral of (dy/dx)= x (x -5)^1/2?

Answer 1

\[(2/15) (-5+x)^{3/2}(10+3x)\]

Answer 2

Using substitution...
\[\Large \begin{array}{l}
\int {x{{(x - 5)}^{1/2}}} dx\\
u = (x - 5)\\
u + 5 = x\\
\frac{{du}}{{dx}} = 1\\
du = 1dx = dx\\
\int {(u + 5)} \cdot {u^{1/2}}du\\
= \int {{u^{3/2}} + 5{u^{1/2}}du} \\
= \int {{u^{3/2}}du} + \int {5{u^{1/2}}} du\\
= \frac{{{u^{5/2}}}}{{5/2}} + \frac{{5{u^{3/2}}}}{{3/2}} + c\\
= \frac{{2{u^{5/2}}}}{5} + \frac{{10{u^{3/2}}}}{3} + c
\end{array}\]
Substitute back in for u and simplify.