Question

Find the following integrals.
$\int \sqrt{x}(3x^2+2x+3)dx.$

Answer 1

$\int \sqrt{x}(3x^2+2x+3)dx=3\int x^{\frac{5}{2}}dx+2\int x^{\frac{3}{2}}dx+3\int x^{\frac{1}{2}}dx=\frac{3.x^{\left(\frac{5}{2}\right)+1}}{\left(\frac{5}{2}\right)+1}+\frac{2.x^{\left(\frac{3}{2}\right)+1}}{\left(\frac{3}{2}\right)+1}+\frac{3.x^{\left(\frac{1}{2}\right)+1}}{\left(\frac{1}{2}\right)+1}+C=\frac{3x^{\frac{7}{2}}}{\frac{7}{2}}+\frac{2x^{\frac{5}{2}}}{\frac{5}{2}}+\frac{3x^{\frac{3}{2}}}{\frac{3}{2}}+C=\frac{6}{7}{x}^{\frac{7}{2}}+\frac{4}{5}{x}^{\frac{5}{2}}+2x^{\frac{3}{2}}+C$