Question

Exemplo Calcule
$\int[\frac {3x+13} {x^2-3x-4}\,\ ] dx$

Answer 1

$\int \frac{3x + 13}{ {x}^{2} - 3x - 4} \: dx = \int \frac{3x + 13}{(x - 4)(x + 1)} \: dx$

$\frac{3x + 13}{(x - 4)(x + 1)} = \frac{A}{x - 4} + \frac{B}{x + 1} \\A = \frac{3.4 + 13}{4 + 1} = \frac{25}{5} = 5 \\ B = \frac{3. (- 1) + 13}{ - 1 - 4} = \frac{10}{ - 5} = - 2$

$\int \frac{3x + 13}{ {x}^{2} - 3x - 4} \: dx \\ = \int \frac{3x + 13}{(x - 4)(x + 1)} \: dx \\ = \int \frac{5}{x - 4}dx + \int \frac{ - 2}{x + 1}dx$

$5 ln|x - 4| - 2 ln|x + 1| + c \\ = ln| { \frac{ {(x - 4)}^{5} }{ {(x + 1)}^{2}}|} + c$