Mathematics

The value of $$\displaystyle\int \dfrac{x^2+5x-1}{\sqrt{x}}dx$$ equals?


SOLUTION

Consider the given integration,

$$ \Rightarrow \int{\dfrac{{{x}^{2}}+5x-1}{\sqrt{x}}}dx $$

$$ \Rightarrow \int{\left( \dfrac{{{x}^{2}}}{\sqrt{x}}+\dfrac{5x}{\sqrt{x}}-\dfrac{1}{\sqrt{x}} \right)}dx $$

$$ \Rightarrow \left( \int{{{x}^{\dfrac{3}{2}}}}+5.{{x}^{\dfrac{1}{2}}}-2{{x}^{-\dfrac{1}{2}}} \right)dx $$

$$ \Rightarrow \dfrac{{{x}^{\dfrac{5}{2}}}}{\dfrac{5}{2}}+5.\dfrac{{{x}^{\dfrac{3}{2}}}}{\dfrac{3}{2}}-\dfrac{{{x}^{\dfrac{1}{2}}}}{\dfrac{1}{2}}+C $$

$$ \Rightarrow \dfrac{2}{5}.{{x}^{\dfrac{5}{2}}}+\dfrac{10}{3}.{{x}^{\dfrac{3}{2}}}-{{x}^{\dfrac{1}{2}}}+C $$

 

Hence, this is the answer.

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