# SOLVE the integration problem

$\begin{array}{l}\text{Solve}\ f\left(x\right)=\int_0^1\left(3x^{\frac{3}{2}}+6x^{\frac{1}{2}}\right)dx\\ f\left(x\right)=\int_0^1\left(3x^{\frac{3}{2}}+6x^{\frac{1}{2}}\right)dx\\ =3\int_0^1\left(x^{\frac{3}{2}}\right)dx+6\int_0^1\left(x^{\frac{1}{2}}\right)dx\\ =3\left(\frac{x^{\frac{3}{2}+1}}{\frac{3}{2}+1}\right)_0^1+6\left(\frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1}\right)_0^1\\ =3\left(\frac{x^{\frac{5}{2}}}{\frac{5}{2}}\right)_{_0}^1+6\left(\frac{x^{\frac{3}{2}}}{\frac{3}{2}}\right)_{_0}^1\\ =3\times\frac{2}{5}\left(1^{\frac{5}{2}}-0^{\frac{5}{2}}\right)+6\times\frac{2}{3}\left(1^{\frac{3}{2}}+0^{\frac{3}{2}}\right)\\ =\frac{6}{5}\times1+4\times1=\frac{26}{5}\\ \int_0^1\left(3x^{\frac{3}{2}}+6x^{\frac{1}{2}}\right)dx=\frac{26}{5}\end{array}$

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