Question

Вычислить неопределенный интеграл

Answer 1

1.

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

2.

$\int\limits \cos(3x - 4) dx = \frac{1}{3} \int\limits \cos(3x - 4) d(3x - 4) = \\ = \frac{1}{3} \sin(3x - 4) + C$

3.

$\int\limits {x}^{5} ln(x) dx \\$

По частям:

$u = ln(x) \: \: \: \: du = \frac{1}{x} dx \\ dv = {x}^{5} dx \: \: v = \frac{ {x}^{6} }{6} \\ \\ uv - \int\limits \: vdu = \\ = \frac{ {x}^{6} }{6} ln(x) - \int\limits \frac{ {x}^{6} }{6} \times \frac{dx}{x} = \\ = \frac{ {x}^{6} }{6} ln(x) - \frac{1}{6} \times \frac{ {x}^{6} }{6} + C = \\ = \frac{ {x}^{6} }{6} ( ln(x) - \frac{1}{6} ) + C$