Задача 43664 ...

sova

2020-01-25 09:57:42

★

x^2+6x+13=x^2+6x+9+4=(x+3)^2+4

Замена
x+3=t
x=t-3

dx=dt



[m]\int \frac{9-4x}{x^2+6x+13}dx=\int \frac{9-4\cdot(t-3)}{t^2+4}dt=\int \frac{21-4t}{t^2+4}dt=[/m]

[m]=21\int \frac{1}{t^2+4}dt-2\int \frac{d(t^2+4)}{t^2+4}=\frac{21}{2}arctg\frac{t}{2}-2ln|t^2+4|+C=[/m]

[m]=10,5arctg\frac{x+3}{2}-2ln|x^2+6x+13|+C[/m]