Question

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Задание на фото ⬆️

Answer 1

$\int_{1}^{4} \frac{ {x}^{2} + x \times {x}^{ \frac{1}{2} } + x }{ {x}^{ \frac{1}{2} } } dx = \int_{1}^{4} \frac{ {x}^{2} + {x}^{ \frac{3}{2} } + x }{ {x}^{ \frac{1}{2} } } dx =$

$\int_{1}^{4} \frac{ {x}^{2} }{ {x}^{ \frac{1}{2} } } + \frac{ {x}^{ \frac{3}{2} } }{ {x}^{ \frac{1}{2} } } + \frac{x}{ {x}^{ \frac{1}{2} } } dx = \int_{1}^{4} {x}^{ \frac{3}{2} } + x + {x}^{ \frac{1}{2} } dx =$

$( \frac{ {x}^{ \frac{5}{2} } }{ \frac{5}{2} } + \frac{ {x}^{2} }{2} + \frac{ {x}^{ \frac{3}{2} } }{ \frac{3}{2} } )| _{1}^{4} = ( \frac{2 {x}^{2} \sqrt{x} }{5} + \frac{ {x}^{2} }{2} + \frac{2x \sqrt{x} }{3} )| _{1}^{4} =$

$\frac{2 \times {4}^{2} \sqrt{4} }{5} + \frac{ {4}^{2} }{2} + \frac{2 \times 4 \sqrt{4} }{3} - ( \frac{2 \times {1}^{2} \sqrt{1} }{5} + \frac{ {1}^{2} }{2} + \frac{2 \times 1 \sqrt{1} }{3} ) =$

$\frac{64}{5} + \frac{16}{2} + \frac{16}{3} - ( \frac{2}{5} + \frac{1}{2} + \frac{2}{3} ) =$

$\frac{64}{5} + \frac{16}{2} + \frac{16}{3} - \frac{2}{5} - \frac{1}{2} - \frac{2}{3} =$

$\frac{62}{5} + \frac{15}{2} + \frac{14}{3} = \frac{372 + 225 + 140}{30} = \frac{737}{30} = 24 \frac{17}{30}$