Question

Интеграл. Помогите решить.​

Answer 1

$\int\limits \frac{e {}^{x} - 1}{1 - 2 {e}^{x} } dx \\ \\ {e}^{x} = t \\e {}^{x} dx = dt \\ dx = \frac{dt}{t} \\ \\ \int\limits \frac{t - 1}{1 - 2t} \times \frac{dt}{t} \\ \\ \frac{t - 1}{t(1 - 2t)} = \frac{a}{t} + \frac{b}{1 - 2t} \\ t - 1 = a(1 - 2t) + bt \\ t - 1 = a - 2at + bt \\ \\ \left \{ {{1 = - 2a + b} \atop { - 1 = a }} \right. \\ \left \{ {{a = - 1 } \atop {b = 1 + 2a = - 1 }} \right. \\ \\ - \int\limits \frac{dt}{t} - \int\limits \frac{dt}{1 - 2t} = \\ = - ln |t| + \frac{1}{2} \int\limits \frac{( - 2)dx}{1 - 2t} = \\ = - ln |t| + \frac{1}{2} \int\limits \frac{d(1 - 2t)}{1 - 2t} = \\ = - ln |t| + \frac{1}{2} ln |1 - 2t| + C= \\ = - ln | {e}^{x}| + \frac{1}{2} ln |1 - 2 {e}^{x} | + C= \\ = - x + \frac{1}{2} ln |1 - 2 {e}^{x} | + C$