Question

Calcule o valor da integral indefinida usando o método para decompor em frações parciais.

teorema a ser usado: ∫$\frac{P(x)}{(x-\alpha)(x-\beta ) }$= A In $|x-\alpha |+BIn|x-\beta |+k$

Answer 1

$\boxed{\begin{array}{l}\sf\dfrac{x}{x^2-4}=\dfrac{x}{(x-2)(x+2)}\\\sf\dfrac{x}{(x-2)(x+2)} =\dfrac{A}{x-2}+\dfrac{B}{x+2}\\\sf A=\dfrac{x}{x+2}\bigg |_{x=2}=\dfrac{2}{2+2}=\dfrac{2}{4}=\dfrac{1}{2}\\\sf B=\dfrac{x}{x-2}\bigg|_{x=-2}=\dfrac{-2}{-2-2}=-\dfrac{2}{-4}=\dfrac{1}{2}\\\displaystyle\sf\int\dfrac{x}{x^2-4}dx=\dfrac{1}{2}\ell n|x-2|+\dfrac{1}{2}\ell n|x+2|+k\\\displaystyle\sf\int\dfrac{x}{x^2-4}dx=\ell n\bigg|[(x-2)(x+2)]^{\frac{1}{2}}\bigg|+k\end{array}}$