Question

integral partes ∫ (x+3)^2 .e^x dx

Answer 1

 Olá Sabrina!

$\int (x + 3)^2 \cdot e^x dx = \\\\\\ \begin{cases} f(x) = (x + 3)^2 \rightarrow f'(x) = 2(x + 3) dx \\ g'(x) = e^x dx \rightarrow g(x) = e^x \end{cases} \\\\\\ \int f(x) \cdot g'(x) dx = f(x) \cdot g(x) - \int f'(x) \cdot g(x) dx \\\\\\ \int (x + 3)^2 \cdot e^x dx = (x + 3)^2 \cdot e^x - \int 2(x + 3) \cdot e^x dx$

 Note que devemos por partes mais uma vez (a última integral)!

 Segue,

$\begin{cases} h(x) = x + 3 \rightarrow h'(x) = 1 dx \\ i'(x) = e^x dx \rightarrow i(x) = e^x \end{cases} \\\\\\ \int (x + 3)^2 \cdot e^x dx = (x + 3)^2 \cdot e^x - 2 \left [ h(x) \cdot i(x) - \int h'(x) \cdot i(x) dx \right ] \\\\\\ \int (x + 3)^2 \cdot e^x dx = (x + 3)^2 \cdot e^x - 2 \left [ (x + 3) \cdot e^x - \int 1 \cdot e^x dx \right ] \\\\\\ \int (x + 3)^2 \cdot e^x dx = (x + 3)^2 \cdot e^x - 2 (x + 3) \cdot e^x + 2 \cdot (e^x + c_1)$

$\int (x + 3)^2 \cdot e^x dx = e^x \left [ (x + 3)^2 - 2(x + 3) + 2 \right ] + \underbrace{2 \cdot c_1}_{C} \\\\\\ \int (x + 3)^2 \cdot e^x dx = e^x (x^2 + 6x + 9 - 2x - 6 + 2) + C \\\\\\ \boxed{\int (x + 3)^2 \cdot e^x dx = e^x (x^2 + 4x + 5) + C}$