Question

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

Answer 1

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

Where $C$ is constant of integration