Question

Qual alternativa representa a resposta correta da seguinte integral definida?
entre(1) e (3) (x²-4x) dx Respostas ( ) 22/3 ( ) 25/9 ( ) 21/9 ( )23/4 ou ( )20/2

Answer 1

$\displaystyle\int\limits_{1}^{3}{(x^{2}-4x)\,dx}\\ \\ \\ =\left.\left(\dfrac{x^{2+1}}{2+1}-4\cdot \dfrac{x^{1+1}}{1+1} \right )\right|_{1}^{3}\\ \\ \\ =\left.\left(\dfrac{x^{3}}{3}-4\cdot \dfrac{x^{2}}{2} \right )\right|_{1}^{3}\\ \\ \\ =\left.\left(\dfrac{x^{3}}{3}-2x^{2} \right )\right|_{1}^{3}\\ \\ \\ =\left(\dfrac{(3)^{3}}{3}-2\cdot (3)^{2} \right )-\left(\dfrac{(1)^{3}}{3}-2\cdot (1)^{2} \right )\\ \\ \\ =\left(9-18 \right )-\left(\dfrac{1}{3}-2 \right )\\ \\ \\ =-9-\left(\dfrac{1-6}{3} \right )\\ \\ \\ =-9-\left(-\dfrac{5}{3} \right )\\ \\ \\ =-9+\dfrac{5}{3}\\ \\ \\ =\dfrac{-27+5}{3}\\ \\ \\ =-\dfrac{22}{3}$


A resposta é $-\dfrac{22}{3},$ com o sinal negativo mesmo.