Question

Resolva, passo a passo a integral definida:

$\int\limits^{2}_{-1} {-x^2+9} \, dx$

Answer 1

Olá

$\displaystyle \int_{-1}^{2}-x^2+9dx\\\\\\\text{Vamos dividir em duas integrais, e colocar o sinal negativo para}\\\text{fora da integral.}\\\\\\-\int_{-1}^{2}x^2dx~+~\int_{-1}^{2}9dx\\\\\\\text{Agora integra pela propriedade: } \int x^pdx~=~ \frac{x^{p+1}}{p+1} \\\\\\= \left.\left(- \frac{x^3}{3} ~+~9x\,\right)\right|_{-1}^{2}\\\\\\\text{Faz a subtracao, limite superior menos limite inferior}\\\\=\left.\left(- \frac{(2)^3}{3} ~+~9\cdot(2)\,\right)-\left.\left(- \frac{(-1)^3}{3} ~+~9\cdot(-1)\,\right)$

$\displaystyle =\left.\left(- \frac{8}{3} ~+~18\,\right)-\left.\left( \frac{1}{3} ~-~9\,\right)$

$\displaystyle =- \frac{8}{3} ~+~18-\frac{1}{3} ~+~9\\\\=-3+27\\\\=\boxed{24}$