Question

calcule a seguinte integral considerando os limites de integração (0, 1): f(x) = x^4 (x^5-2)^5

Answer 1

Calcular a integral definida:

     $\displaystyle\int_0^1 x^4(x^5-2)^5\,dx\\\\\\ =\int_0^1 \frac{1}{5}\cdot 5x^4(x^5-2)^5\,dx\\\\\\ =\frac{1}{5}\int_0^1 (x^5-2)^5\cdot 5x^4\,dx$


Faça a seguinte substituição:

     $x^5-2=u\quad\Rightarrow\quad 5x^4\,dx=du$


Novos limites de integração em $u:$

     $\begin{array}{lcl}\textsf{Quando~~} x=0&\quad\Rightarrow\quad&u=0^5-2\\\\ &&u=0-2\\\\ && u=-2\\\\\\ \textsf{Quando~~} x=1&\quad\Rightarrow\quad&u=1^5-2\\\\ &&u=1-2\\\\ &&u=-1 \end{array}$


Substituindo, a integral fica

     $\displaystyle=\frac{1}{5}\int_{-2}^{-1} u^5\,du\\\\\\ =\frac{1}{5}\cdot \frac{u^{5+1}}{5+1}\bigg|_{-2}^{-1}\\\\\\ =\frac{1}{5}\cdot \frac{u^6}{6}\bigg|_{-2}^{-1}\\\\\\ =\frac{1}{5}\cdot \left(\frac{(-1)^6}{6}-\frac{(-2)^6}{6}\right)\\\\\\ =\frac{1}{5}\cdot \left(\frac{1}{6}-\frac{64}{6}\right)\\\\\\ =\frac{1}{5}\cdot \left(-\,\frac{63}{6}\right)\\\\\\ =-\,\frac{63}{30}\begin{array}{c}^{\div 3}\\ ^{\div 3}\end{array}$

     $=-\,\dfrac{21}{10}\quad\longleftarrow\quad\textsf{resposta.}$


Bons estudos! :-)