Prove
Question:

$\int \sqrt{x}\left(3 x^{2}+2 x+3\right) d x$

Solution:

$\int \sqrt{x}\left(3 x^{2}+2 x+3\right) d x$

$=\int\left(3 x^{\frac{5}{2}}+2 x^{\frac{3}{2}}+3 x^{\frac{1}{2}}\right) d x$

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

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

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

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