Question

Analise as integrais abaixo:

Answer 1

a) $\int{\dfrac{x^{5}+2x-5}{x^{4}}\,dx}$


$=\int{\left[\dfrac{x^{5}}{x^{4}}+\dfrac{2x}{x^{4}}-\dfrac{5}{x^{4}} \right ]\,dx}\\ \\ \\ =\int{\left[x+\dfrac{2}{x^{3}}-\dfrac{5}{x^{4}} \right ]\,dx}\\ \\ \\ =\int{\left[x+2x^{-3}-5x^{-4} \right ]\,dx}\\ \\ \\ =\dfrac{x^{1+1}}{1+1}+\dfrac{2x^{-3+1}}{-3+1}-\dfrac{5x^{-4+1}}{-4+1}+C\\ \\ \\ =\dfrac{x^{2}}{2}+\dfrac{2x^{-2}}{-2}-\dfrac{5x^{-3}}{-3}+C\\ \\ \\ =\dfrac{x^{2}}{2}-x^{-2}+\dfrac{5x^{-3}}{3}+C\\ \\ \\ =\boxed{\begin{array}{c} \dfrac{x^{2}}{2}-\dfrac{1}{x^{2}}+\dfrac{5}{3x^{3}}+C \end{array}}$


b) $\int{\left(\dfrac{x^{4}}{3}-3x^{2}-1 \right )\,dx}$


$=\dfrac{1}{3}\cdot \dfrac{x^{4+1}}{4+1}-\dfrac{3x^{2+1}}{2+1}-x+C\\ \\ \\ =\dfrac{1}{3}\cdot \dfrac{x^{5}}{5}-\dfrac{3x^{3}}{3}-x+C\\ \\ \\ =\boxed{\begin{array}{c} \dfrac{x^{5}}{15}-x^{3}-x+C \end{array}}$