Question

Find the following integrals.
$\int (1-x)\sqrt{x}dx.$

Answer 1

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