Пр.2

$$ \int \frac{dt}{t^2+2t} = \int \frac{Adt}{t} + \int \frac{Bdt}{t+2} \\

\frac{1}{t^2+2t} = \frac{A}{t} + \frac{B}{t+2} \\

1 = A(t+2)+B(t) \\

1 = At+2A+Bt \\

(A+B)t + 2A-1 = 0 \\

2A-1 = 0, A=\frac{1}{2}, A+B=0, B=-A = -\frac{1}{2} \\

\int \frac{dt}{t^2+2t} = \int\frac{dt}{2dt}-\int\frac{dt}{2(t+2)} = \\

= \frac{1}{2} ln|t| - \frac{1}{2} ln|t+2|+C $$M