Question

integral impropria dx/9+x² de -∞ até +∞

Answer 1

Resolução da questão, veja:

Vamos utilizar a seguinte propriedade para resolvermos isso:

$\Large\boxed{\boxed{\boxed{\boxed{\mathbf{\dispalystyle\int\limits_{a}^{\infty}~f(t)~dt=\displaystyle\lim_{u~\to~\infty}\displaystyle\int\limits_{a}^{u}~f(t)~dt}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}$

Veja:

$\mathtt{\displaystyle\int\limits_{-\infty}^{+\infty}~\dfrac{dx}{9+x^{2}}~dd}}}}}}\\\\\\\\ \mathtt{\displaystyle\lim_{t~\to~-\infty}\displaystyle\int\limits_{t}^{t}~\dfrac{dx}{9+x^{2}}~dd}}}}\\\\\\\\ \mathtt{\dfrac{x}{9+x^{2}}~\displaystyle\int\limits_{1}^{t}~dd~d}}}}}}}\\\\\\\\ \mathtt{\dfrac{x}{9+x^{2}}}~\mathtt{\bigg(\dfrac{d^{2}}{2}-\dfrac{d^{2}}{2}\bigg)}}}}}}}}}}}}}}}\\\\\\\\\ \mathtt{\dfrac{x}{9+x^{2}}}~\mathtt{\bigg(\dfrac{t^{2}}{2}-\dfrac{t^{2}}{2}\bigg)}}}}}}}}}}}}}}}\\\\\\\\\ \mathtt{\dfrac{x}{9+x^{2}}~\cdot~\mathtt{0}}}}}\\\\\\\\ \mathtt{0}}}}\\\\\\\\ \mathtt{\displaystyle\lim_{t~\to~-\infty}0}}}}}}\\\\\\\\\\ \Large\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\mathbf{~\therefore~\displaystyle\int\limits_{-\infty}^{+\infty}~\dfrac{dx}{9+x^{2}}~dd=0.}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}$

Espero que te ajude. '-'