Question
Evaluate the following integrals:
$\int\frac{(1+\text{x})^3}{\sqrt{\text{x}}}\text{dx}$
$\int\frac{(1+\text{x})^3}{\sqrt{\text{x}}}\text{dx}$
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Answer
$\int\frac{(1+\text{x})^3}{\sqrt{\text{x}}}\text{dx}$
$=\int\frac{1}{\sqrt{\text{x}}}\text{dx}+\int\frac{\text{x}^3}{\sqrt{\text{x}}}\text{dx}+\int\frac{3\text{x}^2}{\sqrt{\text{x}}}\text{dx}+\int\frac{3\text{x}}{\sqrt{\text{x}}}\text{dx}$
$=\int\text{x}^{\frac{-1}{2}}\text{dx}+\int\text{x}^{\frac{5}{2}}\text{dx}+3\int\text{x}^{\frac{3}{2}}\text{dx}+3\int\text{x}^{\frac{1}{2}}\text{dx}$
$=\frac{\text{x}^{\frac{-1}{2}+1}}{\frac{-1}{2}+1}+\frac{\text{x}^{\frac{5}{2}+1}}{\frac{5}{2}+1}+\frac{3\text{x}^{\frac{3}{2}+1}}{\frac{3}{2}+1}+\frac{3\text{x}^{\frac{1}{2}+1}}{\frac{1}{2}+1}+\text{C}$
$=\frac{\text{x}^{\frac{1}{2}}}{\frac{1}{2}}+\frac{\text{x}^{\frac{7}{2}}}{\frac{7}{2}}+\frac{\text{3x}^{\frac{5}{2}}}{\frac{5}{2}}+\frac{\text{3x}^{\frac{3}{2}}}{\frac{3}{2}}+\text{C}$
$=2\text{x}^{\frac{1}{2}}+\frac{2}{7}\text{x}^{\frac{7}{2}}+\frac{6}{5}\text{x}^{\frac{5}{2}}+\frac{6}{3}\text{x}^{\frac{3}{2}}+\text{C}$
$=2\text{x}^{\frac{1}{2}}+\frac{2}{7}\text{x}^{\frac{7}{2}}+\frac{6}{5}\text{x}^{\frac{5}{2}}+2\text{x}^{\frac{3}{2}}+\text{C}$
$=\int\frac{1}{\sqrt{\text{x}}}\text{dx}+\int\frac{\text{x}^3}{\sqrt{\text{x}}}\text{dx}+\int\frac{3\text{x}^2}{\sqrt{\text{x}}}\text{dx}+\int\frac{3\text{x}}{\sqrt{\text{x}}}\text{dx}$
$=\int\text{x}^{\frac{-1}{2}}\text{dx}+\int\text{x}^{\frac{5}{2}}\text{dx}+3\int\text{x}^{\frac{3}{2}}\text{dx}+3\int\text{x}^{\frac{1}{2}}\text{dx}$
$=\frac{\text{x}^{\frac{-1}{2}+1}}{\frac{-1}{2}+1}+\frac{\text{x}^{\frac{5}{2}+1}}{\frac{5}{2}+1}+\frac{3\text{x}^{\frac{3}{2}+1}}{\frac{3}{2}+1}+\frac{3\text{x}^{\frac{1}{2}+1}}{\frac{1}{2}+1}+\text{C}$
$=\frac{\text{x}^{\frac{1}{2}}}{\frac{1}{2}}+\frac{\text{x}^{\frac{7}{2}}}{\frac{7}{2}}+\frac{\text{3x}^{\frac{5}{2}}}{\frac{5}{2}}+\frac{\text{3x}^{\frac{3}{2}}}{\frac{3}{2}}+\text{C}$
$=2\text{x}^{\frac{1}{2}}+\frac{2}{7}\text{x}^{\frac{7}{2}}+\frac{6}{5}\text{x}^{\frac{5}{2}}+\frac{6}{3}\text{x}^{\frac{3}{2}}+\text{C}$
$=2\text{x}^{\frac{1}{2}}+\frac{2}{7}\text{x}^{\frac{7}{2}}+\frac{6}{5}\text{x}^{\frac{5}{2}}+2\text{x}^{\frac{3}{2}}+\text{C}$
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