Question

alguem pode me ajudar na integral definida ?

Answer 1

$\displaystyle\int_a^b (Px^2+Qx+R)\,dx$


Regra para encontrar primitivas de potências:

$\displaystyle\int x^n\,dx=\dfrac{x^{n+1}}{n+1}\,,~~~~\text{com }n\ne -1.$


Primitivando a função da integral, e avaliando nos extremos:

$\displaystyle\int_a^b (Px^2+Qx+R)\,dx\\\\\\ =\left.\left(\dfrac{Px^{2+1}}{{2+1}}+\dfrac{Qx^{1+1}}{1+1}+Rx\right)\right|_a^b\\\\\\ =\left.\left(\dfrac{Px^3}{3}+\dfrac{Qx^2}{2}+Rx\right)\right|_a^b\\\\\\ =\left(\dfrac{P\cdot b^3}{3}+\dfrac{Q\cdot b^2}{2}+R\cdot b\right)-\left(\dfrac{P\cdot a^3}{3}+\dfrac{Q\cdot a^2}{2}+R\cdot a\right)\\\\\\ =\dfrac{P\cdot (b^3-a^3)}{3}+\dfrac{Q\cdot (b^2-a^2)}{2}+R\cdot (b-a)\\\\\\\\ \therefore~~\boxed{\begin{array}{c}\displaystyle\int_a^b (Px^2+Qx+R)\,dx=\dfrac{P\cdot (b^3-a^3)}{3}+\dfrac{Q\cdot (b^2-a^2)}{2}+R\cdot (b-a) \end{array}}$