Question

integral de (2x ^2 + 2x - 3)^10 * (2x + 1) dx

Answer 1

Calcular a integral indefinida

     $\displaystyle\int (2x^2+2x-3)^{10}\cdot (2x+1)\,dx\\\\\\ =\int (2x^2+2x-3)^{10}\cdot \frac{1}{2}\cdot 2(2x+1)\,dx\\\\\\ =\frac{1}{2}\int (2x^2+2x-3)^{10}\cdot (4x+2)\,dx$


Faça a seguinte substituição:

     $\begin{array}{lcl}u=2x^2+2x-3&\quad\Rightarrow\quad&du=\Big(2\cdot (2x^{2-1})+2+0\Big)\,dx\\\\&& du=(4x+2)\,dx\end{array}$


Substituindo, a integral fica

     $\displaystyle=\frac{1}{2}\int u^{10}\,du$


Aplique a regra para integrar potências, e a integral fica

     $\displaystyle=\frac{1}{2}\cdot \frac{u^{10+1}}{10+1}+C\\\\\\ =\frac{1}{2}\cdot \frac{u^{11}}{11}+C\\\\\\ =\frac{1}{22}\,u^{11}+C$

     $=\dfrac{1}{22}\,(2x^2+2x-3)^{11}+C$    <————    esta é a resposta.


Bons estudos! :-)