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Question
\[\int\left\{ \sqrt{x}\left( a x^2 + bx + c \right) \right\} dx\]
Sum
Solution
\[\int\sqrt{x} \left( a x^2 + bx + c \right)dx\]
\[ = \int x^\frac{1}{2} \left( a x^2 + bx + c \right)dx\]
`=∫ (ax^{2 + 1/2} + bx^{1/2+1 }+ c x^{1/2})dx`
`= a∫ x^{5/2 }dx + b∫ x^{3/2 }dx + c∫ x^{1/2 } dx`
`= a [ x^(5/2 + 1)/(5/2+ 1) ]+ b[ x^(3/2+1)/(3/2+ 1) ] + c[ x^(1/2+1)/(1/2 + 1 )]+ C`
\[ = \frac{2a}{7} x^\frac{7}{2} + \frac{2b}{5} x^\frac{3}{2} + \frac{2c}{3} x^\frac{3}{2} + C\]
\[ = \int x^\frac{1}{2} \left( a x^2 + bx + c \right)dx\]
`=∫ (ax^{2 + 1/2} + bx^{1/2+1 }+ c x^{1/2})dx`
`= a∫ x^{5/2 }dx + b∫ x^{3/2 }dx + c∫ x^{1/2 } dx`
`= a [ x^(5/2 + 1)/(5/2+ 1) ]+ b[ x^(3/2+1)/(3/2+ 1) ] + c[ x^(1/2+1)/(1/2 + 1 )]+ C`
\[ = \frac{2a}{7} x^\frac{7}{2} + \frac{2b}{5} x^\frac{3}{2} + \frac{2c}{3} x^\frac{3}{2} + C\]
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