Question

Срочно помогите пожалуйста!!!!
Даю 50 балов

Answer 1

1

$f(x) = \int\limits {x}^{7} dx = \frac{ {x}^{8} }{8} + C$

2

$f(x) = \int\limits {(3x - 1)}^{7} dx = \\ = \frac{1}{3} \int\limits {(3x - 1)}^{7} d(3x - 1) = \\ = \frac{1}{3} \times \frac{ {(3x - 1)}^{8} }{8} +C = \frac{ {(3x - 1)}^{8} }{24} + C$

3.

$f(x) = \int\limits(6 {x}^{2} - 3x + 1)dx = \\ = \frac{6 {x}^{3} }{3} - \frac{3 {x}^{2} }{2} + x + C = \\ = 2 {x}^{3} - 1.5 {x}^{2} + x + C$

4.

$f(x) = \int\limits( \frac{3}{ {x}^{5 } } + \frac{4 }{ {x}^{3} } + 2)dx = \\ = \int\limits(3 {x}^{ - 5} + 4 {x}^{ - 3} + 2)dx = \\ = \frac{3 {x}^{ - 4} }{ - 4} + \frac{4 {x}^{ - 2} }{ - 2} + 2x + C = \\ = - \frac{3}{4 {x}^{4} } - \frac{2}{ {x}^{2} } + 2x + C$

5.

$f(x) = \int\limits \sin(3x - \frac{\pi}{3} ) = \\ = \frac{1}{3} \int\limits \sin(3x - \frac{\pi}{6} ) d(3x - \frac{\pi}{3} ) = \\ = - \frac{1}{3} \cos(3x - \frac{\pi}{3} ) + C$

6.

$f(x) = \int\limits( \sin {}^{2} (x) + \cos {}^{2} (x)) dx =\int\limits1 \times dx = x + C$

7.

$f(x) = \int\limits( \cos {}^{2} ( \frac{x}{2} ) - \sin {}^{2} ( \frac{x}{2} ) )dx = \\ = \int\limits \cos(x) dx = \sin(x) + C$

8

$f(x) = \int\limits \frac{2}{ \cos{}^{2} ( \frac{x}{4} - 1 ) } dx = \\ = 2 \times 4\int\limits \frac{d( \frac{x}{4} - 1) }{ \cos {}^{2} ( \frac{x}{4} - 1) } = \\ = 8tg( \frac{x}{4} - 1) + C$