Question

Evaluate : $\int\limits^1_0 {\frac{ x^{2}+3x+2}{ \sqrt{x}}} \, dx$

Answer 1

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Answer 2

Answer:

$\limits^1_0\int\:\frac{x^{2} +3x+2}{\sqrt{x} } dx=\frac{32}{5}$

Step-by-step explanation:

$\limits^1_0\int\:\frac{x^{2} +3x+2}{\sqrt{x} } dx\\ \\ \\ =\int\:\frac{x^{2} }{\sqrt{x} }dx+\int\:\frac{3x}{\sqrt{x} }dx+\int\:\frac{2}{\sqrt{x} }dx\\ \\ \\ =\int\:x^{\frac{3}{2} }dx+\int\:3\sqrt{x}\:dx+\int\:2\:x^{\frac{-1}{2} }dx$

by power rule we can integrate the given function

$=\frac{2\:x^{\frac{5}{2} } }{5} +\frac{6\:x^{\frac{3}{2} } }{3}+4\sqrt{x}+C\\\\\\=\frac{2\:x^{\frac{5}{2} } }{5} +2\:x^{\frac{3}{2} } +4\sqrt{x}+C$

Now put the limits

$=\frac{2\:(1)^{\frac{5}{2} } }{5} +2\:(1)^{\frac{3}{2} } +4\sqrt{1}+\frac{2\:(0)^{\frac{5}{2} } }{5} +2\:(0)^{\frac{3}{2} } +4\sqrt{0}\\ \\ \\=\frac{2}{5}+2+4\\ \\=\frac{2}{5}+6\\ \\ \\ =\frac{32}{5}$