Question

Integral mudança de variável
|(2x^2+2x-3)^10×(2x+1)dx

Answer 1

Calcular a integral indefinida


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


Faça a seguinte mudança de variável:

$2x^2+2x-3=u\quad\Rightarrow\quad (4x+2)\,dx=du$

e a integral fica

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


Use a regra para integrar potências:

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

$=\dfrac{1}{22}\,(2x^2+2x-3)^{11}+C \quad\longleftarrow\quad\textsf{resposta.}$

Bons estudos! :-)