Question

O cálculo da integral resulta em?

Answer 1

$\int_{0}^{5}{\dfrac{dx}{\sqrt[5]{x}}}\\ \\ \\ =\int_{0}^{5}{\dfrac{dx}{x^{1/5}}}\\ \\ \\ =\int_{0}^{5}{x^{-1/5}\,dx}\\ \\ \\ =\left[\dfrac{x^{-1/5+1}}{-1/5+1} \right ]_{0}^{5}\\ \\ \\ =\left[\dfrac{x^{4/5}}{4/5} \right ]_{0}^{5}\\ \\ \\ =\dfrac{5}{4}\cdot \left[x^{4/5} \right ]_{0}^{5}\\ \\ \\ =\dfrac{5}{4}\cdot \left[\sqrt[5]{x^{4}} \right ]_{0}^{5}\\ \\ \\ =\dfrac{5}{4}\cdot \left[\sqrt[5]{5^{4}}-\sqrt[5]{0^{4}} \right ]\\ \\ \\ =\dfrac{5}{4}\cdot \left[\sqrt[5]{625}-0 \right ]\\ \\ \\ =\dfrac{5\sqrt[5]{625}}{4}\\ \\ \\ \boxed{ \int_{0}^{5}{\dfrac{dx}{\sqrt[5]{x}}}=\dfrac{5\sqrt[5]{625}}{4} }$