Question

Integral da x²*e elevado a 3x dx

Answer 1

Calcular a integral indefinida

     $\displaystyle\int x^2 e^{3x}\,dx$


Usaremos o método de integração por partes:

     $\begin{array}{lcl} u=x^2&\quad\Rightarrow\quad&du=2x\,dx\\\\ dv=e^{3x}\,dx&\quad\Leftarrow\quad&v=\dfrac{1}{3}\,e^{3x} \end{array}$


     $\displaystyle\int u\,dv=uv-\int v\,du\\\\\\ \int x^2 e^{3x}\,dx=x^2\cdot \frac{1}{3}\,e^{3x}-\int \frac{1}{3}\,e^{3x}\cdot 2x\,dx\\\\\\ \int x^2 e^{3x}\,dx=\frac{1}{3}\,x^2e^{3x}-\frac{2}{3}\int xe^{3x}\,dx\qquad\quad\mathbf{(i)}$


Para computar a integral que apareceu no lado direito, usamos integração por partes novamente:

     $\begin{array}{lcl} u=x&\quad\Rightarrow\quad&du=dx\\\\ dv=e^{3x}\,dx&\quad\Leftarrow\quad&v=\dfrac{1}{3}\,e^{3x} \end{array}$


     $\displaystyle\int u\,dv=uv-\int v\,du\\\\\\ \int x e^{3x}\,dx=x\cdot \frac{1}{3}\,e^{3x}-\int \frac{1}{3}\,e^{3x}\, dx\\\\\\ \int xe^{3x}\,dx=\frac{1}{3}\,xe^{3x}-\frac{1}{3}\int e^{3x}\,dx\\\\\\ \int xe^{3x}\,dx=\frac{1}{3}\,xe^{3x}-\frac{1}{3}\cdot \left(\frac{1}{3}\,e^{3x}\right)\\\\\\ \int xe^{3x}\,dx=\frac{1}{3}\,xe^{3x}-\frac{1}{9}\,e^{3x}\qquad\quad \mathbf{(ii)}$

     (constante de integração omitida propositalmente)


Substituindo  (ii)  em  (i),  obtemos

     $\displaystyle\int x^2 e^{3x}\,dx=\frac{1}{3}\,x^2e^{3x}-\frac{2}{3}\cdot \left(\frac{1}{3}\,xe^{3x}-\frac{1}{9}\,e^{3x}\right)+C\\\\\\ \int x^2 e^{3x}\,dx=\frac{1}{3}\,x^2e^{3x}-\frac{2}{9}\,xe^{3x}+\frac{2}{27}\,e^{3x}+C\\\\\\ \int x^2 e^{3x}\,dx=\frac{9}{27}\,x^2e^{3x}-\frac{6}{27}\,xe^{3x}+\frac{2}{27}\,e^{3x}+C$

     $\displaystyle\int x^2 e^{3x}\,dx=\frac{1}{27}\,e^{3x}\cdot (9x^2-6x+2)+C\quad\longleftarrow\quad\textsf{esta \'e a resposta.}$


Bons estudos! :-)