Find the integral: x (3 x^ 2 +2 x+3 ) d x - Vidyadip

Question

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

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Answer

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

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