Question

O valor da integral definida ∫⁰₋₃ $\frac{2x⁴ - 8x³ + 14x²}{2x²}$ dx é:


a) 36

b) 14

c) 0

d) 48

e) 29

Answer 1

Primeiro efetuarei a divisão : 

(2x^4 - 8x³ + 14x²)/2x² = x² - 4x + 7 


Agora resolvo minha integral : 



$\int\limits^0_{-3} {x^{2}-4x+7} \, dx \\\\\\ \frac{x^{2+1}}{2+1} - \frac{4x^{1+1}}{1+1} + \frac{7x^{0+1}}{0+1} \\\\\\ \frac{x^{3}}{3} - \frac{4x^{2}}{2} + \frac{7x}{1} \\\\\\\boxed{ \frac{x^{3}}{3} -2x^{2}+7x+c}$



Substituindo os valores (superior - inferior) ... 



$| \frac{x^{3}}{3} -2x^{2}+7x|-| \frac{x^{3}}{3} -2x^{2}+7x|\\\\\\| \frac{0^3}{3} -2.0^{2}+7.0|-| \frac{(-3)^{3}}{3} -2.(-3)^{2}+7.(-3)|\\\\\\| \frac{0}{3} -2.0+0|-| \frac{-27}{3} -2.9-21|\\\\\\|0-0+0|-|-9-18-21|\\\\\\|0|-|-48|\\\\\\0+48=\boxed{\boxed{48}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Letra\ d) \ \ \ \ \ \ \ \ \ \ \ \ ok$