Question

integral d ∫dx/(4+x²) ?

Answer 1

corrigido solução na imagem.

Answer 2

Integrais que produzem funções trigonométricas inversas

$\displaystyle\mathsf{\int\dfrac{dt}{\sqrt{a^2-t^2}}=arcsen(\dfrac{t}{a})+k}$

$\displaystyle\mathsf{\int\dfrac{dt}{t^2+a^2}=\dfrac{1}{a}arctg(\dfrac{t}{a})+k}$

$\displaystyle\mathsf{\int\dfrac{dt}{t\sqrt{t^2-a^2}}=\dfrac{1}{a}arcsec|\dfrac{t}{a}|+k}$

$\dotfill$

$\displaystyle\mathsf{\int\dfrac{dx}{4+x^2}}$

faça

$\mathsf{a^2=4\implies~a=\sqrt{4}=2}\\\mathsf{t^2=x^2\implies~t=x}\\\mathsf{dt=dx}$

$\displaystyle\mathsf{\int\dfrac{dx}{x^2+4}=\int\dfrac{dt}{t^2+a^2}}\\\mathsf{\dfrac{1}{a}arctg(\dfrac{t}{a})+k}\\\displaystyle\mathsf{\int\dfrac{dx}{x^2+4}=\dfrac{1}{2}arctg(\dfrac{x}{2})+k}$