Question

Найти неопределенный интеграл

Answer 1

$\int\limits \frac{dx}{5x - {x}^{2} - 6} \\ \\ 5x - {x}^{2} - 6 = - ( {x}^{2} - 5x + 6) = \\ = - ( {x}^{2} - 2 \times x \times \frac{5}{2} + \frac{25}{4} - \frac{1}{4} ) = \\ = - ((x - \frac{5}{2} ) {}^{2} -( \frac{1}{2} ) {}^{2} ) = ( \frac{1}{2} ) {}^{2} - (x - \frac{5}{2} ) {}^{2} \\ \\ \int\limits \frac{d(x - \frac{5}{2}) }{( \frac{1}{2}) - (x - \frac{5}{2}) {}^{2} } = \frac{1}{2 \times \frac{1}{2} } ln( | \frac{ \frac{1}{2} + x - \frac{5}{2} }{ \frac{1}{2} - x + \frac{5}{2} } | ) + C = \\ = ln( | \frac{x - 2}{3 - x} | ) + C$