Question

O valor da integral definida 1 a 0 x/x ao quadrado +1 dx é:
0
1/2 1n2
1/2
1
2

Answer 1

Resolução da questão, vejamos:

Vamos substituir u = x² por x = 2u, teremos: du = 2xdx, veja na prática:

$\mathtt{\displaystyle\int\limits_{1}^{0}~\dfrac{x}{x^{2}+1}}~\mathtt{dx}}}}\\\\\\\\\ \mathtt{\displaystyle\int\limits_{0^{2}}^{1^{2}}~\dfrac{1}{2u+2}}~\mathtt{du}}}\\\\\\\\ \mathtt{\displaystyle\int\limits_{0}^{1}~\dfrac{1}{2u+2}}~\mathtt{du}}}\\\\\\\\\ \mathtt{\displaystyle\int\limits_{0}^{1}~\dfrac{1}{2(u+2)}}~\mathtt{du}}}~~\textsf{Puxando~as~constantes~para~fora,~teremos}:\\\\\\\\\ \mathtt{\dfrac{1}{2}~ln~2.}}\\\\\\\\\ \Large\boxed{\boxed{\boxed{\boxed{\mathbf{~\therefore~\displaystyle\int\limits_{1}^{0}~\dfrac{x}{x^{2}+1}}~\mathbf{dx}=\dfrac{1}{2}~ln~2.}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}$

Alternativa “B” é a correta.

Espero que te ajude. '-'