Question

решить неопределенный интеграл​

Answer 1

1

$\int\limits \frac{dx}{9 {x}^{2} - 1 } = \int\limits \frac{dx}{ {(3x)}^{2} - 1 } = \\ = \frac{1}{3} \int\limits \frac{d(3x)}{ {(3x)}^{2} - {1}^{2} } = \frac{1}{3} \times \frac{1}{2 \times 1} ln | \frac{3x - 1}{3 x + 1} | + C = \\ = \frac{1}{6} ln | \frac{3x - 1}{3x + 1} | + C$

2

$\int\limits \cos( \frac{2x}{3} ) dx = \frac{3}{2} \int\limits \cos( \frac{2x}{3} ) d( \frac{2x}{3} ) = \frac{3}{2} \sin( \frac{3x}{2} ) + C \\$

3

$\int\limits\frac{dx}{5x + 3} = \frac{1}{5} \int\limits \frac{d(5x)}{5x + 3} = \\ = \frac{1}{5} \int\limits \frac{d(5x) + 3)}{5x + 3} = \frac{1}{5} ln |5x + 3| + C$

4

$\int\limits \frac{dx}{x \sqrt{ ln(x) } } \\ \\ ln(x) = t \\ \frac{dx}{x} = dt \\ \\ \int\limits \frac{dt}{ \sqrt{t} } = \int\limits {t}^{ - \frac{1}{2} } dt = \frac{ {t}^{ \frac{1}{2} } }{ \frac{1}{2} } + C = \\ = 2 \sqrt{t} + C = 2 \sqrt{ ln(x) } + c$