Question

6) Resolva a integral e diga o método que está usando:

$\int\limits { \frac{x-9}{(x+5)(x-2)} } \, dx$

Answer 1

Resolução da questão, vejamos:

Vamos reescrever essa integral em frações parciais, observe:

$\mathsf{\displaystyle\int~\dfrac{x-9}{(x+5)(x-2)}~dx}}}=\mathsf{\displaystyle\int~\dfrac{2}{(x+5)}~dx-\displaystyle\int~\dfrac{1}{(x-2)}~dx}}}}}}\\\\\\\\ \mathsf{2~\cdot~\displaystyle\int~\dfrac{1}{(x+5)}~dx-\displaystyle\int~\dfrac{1}{(x-2)}~dx}}}}}}$

Para prosseguirmos com a resolução vamos utilizar a seguinte regra de integração:

$\Large{\boxed{\boxed{\mathbf{\displaystyle\int~\dfrac{1}{x+a}~dx}=\mathrm{ln}~|\mathbf{x+a}|}}}}}}}}}}}}}}}}$

Portanto, por dedução temos que:

$\mathsf{2~\cdot~\displaystyle\int~\dfrac{1}{(x+5)}~dx-\displaystyle\int~\dfrac{1}{(x-2)}dx}} =\mathsf{2~\cdot~\textrm{ln}|x+5|-|x-2|+C}}}}\\\\\\\\\ \Large{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\mathbf{~\therefore~\displaystyle\int~\dfrac{x-9}{(x+5)(x-2)}}~\mathbf{dx}=\mathbf{2~\cdot~\textrm{ln}|x+5|-|x-2|+C}}}}}}}}}}}}}}}}}}}}}}}}} ~~\checkmark}}$

Espero que te ajude. '-'

Answer 2

$\mathsf{\dfrac{x-9}{(x+5)(x-2)} =\dfrac{A}{x+5}+\dfrac{B}{x-2}}\\\mathsf{A=\dfrac{-5-9}{-5-2}=-\dfrac{14}{-7}=2}\\\mathsf{B=\dfrac{2-9}{2+5}=-\dfrac{7}{7}=-1}$

$\displaystyle\mathsf{\int\dfrac{x-9}{(x+5)(x-2)}dx=2\int\dfrac{dx}{x+5}-1\int\dfrac{dx}{x+2}}\\\boxed{\boxed{\displaystyle\mathsf{\int\dfrac{x-9}{(x+5)(x-2)}dx=\large\ell n\left|\dfrac{(x+5)^2}{x+2}\right|+k}}}$