Question

Calcule a seguinte integral indefinida

Answer 1

Resposta:

$2ln\frac{|x-2|}{|x-3|}+c$

Explicação passo a passo:

$\frac{-2}{x^2-5x+6} =\frac{A}{x-2}+\frac{B}{x-3}=\frac{A(x-3)+B(x-2)}{(x-2)(x-3)} \implies\impliesa A(x-3)+B(x-2)=-2\\\\Para~~x=3\implies A.0+B(3-2)=-2 \implies B=-2\\\\Para~~x=2 \implies A(2-3)+B.0=-2\implies-A=-2\implies A=2\\\\\frac{-2}{x^2-5x+6} =\frac{2}{x-2}+\frac{-2}{x-3}=\frac{2}{x-2}-\frac{2}{x-3} \\\\ \displaystyle\int\frac{-2}{x^2-5x+6}dx= \displaystyle\int(\frac{2}{x-2} -\frac{2}{x-3})dx=2 \displaystyle\int\frac{dx}{x-2} -2 \displaystyle\int\frac{dx}{x-3}=\\\\2ln|x-2|-2ln|x-3| +c$