Question

Find the integral: $\int (1-x)\sqrt{x}dx$

Answer 1

$\int (1-x)\sqrt{x}dx$
$=\int (x^{\frac{1}{2}}-x^{1+\frac{1}{2}})dx$
$=\int (x^{\frac{1}{2}}-x^{\frac{3}{2}})dx$
$=\int x^{\frac{1}{2}}dx-\int x^{\frac{3}{2}}dx$
$=\left(\frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1}\right)-\left(\frac{x^{\frac{3}{2}+1}}{\frac{3}{2}+1}\right)+C$
$=\left(\frac{x^{\frac{3}{2}}}{\frac{3}{2}}\right)-\left(\frac{x^{\frac{5}{2}}}{\frac{5}{2}}\right)+C$
$=\frac{2x^{\frac{3}{2}}}{3}-\frac{2x^{\frac{5}{2}}}{5}+C$

Where $C$ is constant of integration