Question

integral de (x+1) raiz quadrada de 2x+x^2 dx​

Answer 1

Resposta:    $\displaystyle\int (x+1)\sqrt{2x+x^2}\,dx=\frac{1}{3}\sqrt{(2x+x^2)^3}+C.$

Explicação passo a passo:

Calcular a integral indefinida

    $\displaystyle\int (x+1)\sqrt{2x+x^2}\,dx\\\\\\=\frac{1}{2}\cdot \int 2(x+1)\sqrt{2x+x^2}\,dx\\\\\\=\frac{1}{2}\int (2x+2)\sqrt{2x+x^2}\,dx\\\\\\ =\frac{1}{2}\int \sqrt{2x+x^2}\cdot (2+2x)\,dx\qquad\mathrm{(i)}$

Faça a seguinte substituição:

    $2x+x^2=u\quad\Longrightarrow\quad (2+2x)\,dx=du$

Substituindo, a integral (i) fica

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

Integre aplicando a regra para primitiva de potências:

    $=\dfrac{1}{2}\cdot \dfrac{u^{(1/2)+1}}{\frac{1}{2}+1}+C\\\\\\ =\dfrac{1}{2}\cdot \dfrac{u^{3/2}}{\frac{3}{2}}+C\\\\\\ =\dfrac{1}{\diagup\!\!\!\! 2}\cdot \dfrac{\diagup\!\!\!\! 2}{3}\,u^{3/2}+C\\\\\\ =\dfrac{1}{3}\,u^{3/2}+C$

Voltando à variável x:

    $=\dfrac{1}{3}\,(2x+x^2)^{3/2}+C\\\\\\=\dfrac{1}{3}\sqrt{(2x+x^2)^3}+C\quad\longleftarrow\quad\mathsf{resposta.}$

Dúvidas? Comente.

Bons estudos!