Question

Find the following integral: $\int \sqrt x(3x^2+2x+3)dx$

Answer 1

$\int \sqrt x(3x^2+2x+3)dx$

= $\int \bigg(3x^{\frac{5}{2}}+2x^{\frac{3}{2}}+3x^{\frac{1}{2}}\bigg)dx$

=$3 \int x^{\frac{5}{2}}dx+2 \int x^{\frac{3}{2}}dx+3 \int x^{\frac{1}{2}}dx$

=$3\bigg(\frac{x^{\frac{7}{2}}}{\frac{7}{2}}\bigg)+2\bigg(\frac{x^{\frac{5}{2}}}{\frac{5}{2}}\bigg)+3\frac{x^{\frac{3}{2}}}{\frac{3}{2}}+C$

=$\frac{6}{7}x^{\frac{7}{2}}+\frac{4}{5}x^{\frac{5}{2}}+2x^{\frac{3}{2}}+C$