Question

Find the integral of ∫(y-5) dy ∕ y^2 + 5y + 6

Answer 1

$$\int \frac {y - 5}{y ^ {2} + 5 y + 6} d y = \dots$$ $$\frac {y - 5}{y ^ {2} + 5 y + 6} = \frac {y - 5}{(y + 2) (y + 3)} = \frac {A}{y + 2} + \frac {B}{y + 3} = \frac {(A + B) y + (3 A + 2 B)}{(y + 2) (y + 3)}$$ $$\left\{ \begin{array}{l} A + B = 1 \\ 3 A + 2 B = - 5 \end{array} \right.$$ $$\left\{ \begin{array}{l} 2 A + 2 B = 2 \\ 3 A + 2 B = - 5 \end{array} \right.$$ $$- A = 7$$ $$A = - 7, B = 8$$ $$\frac {y - 5}{y ^ {2} + 5 y + 6} = \frac {- 7}{y + 2} + \frac {8}{y + 3}$$ $$\ldots = \int \left(\frac {- 7}{y + 2} + \frac {8}{y + 3}\right) d y = - 7 \ln | y + 2 | + 8 \ln | y + 3 | + C, C = c o n s t$$