Question

Помогите срочно! Пожалуйста ​

Answer 1

Ответ:

1.

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

2.

$F(x) = \int\limits( \frac{1}{x} - \frac{3}{ {x}^{3} } )dx = \int\limits( \frac{1}{x} - 3 {x}^{ - 3} )dx = \\ = ln( |x| ) - \frac{3 {x}^{ - 2} }{( - 2)} + C = ln( |x| ) + \frac{3}{2 {x}^{2} } + c$

3.

$F(x) = \int\limits ({x}^{5} - 2x)dx = \frac{ {x}^{6} }{6} - \frac{2 {x}^{2} }{2} + C = \\ = \frac{ {x}^{6} }{6} + {x}^{2} + C$

4.

$F(x) = \int\limits( - \frac{3}{ {x}^{2} } + \frac{4}{ {x}^{3} } )dx =\int\limits ( - 3 {x}^{ - 2} + 4 {x}^{ - 3}) dx = \\ = \frac{ - 3 {x}^{ - 1} }{ - 1} + \frac{4 {x}^{ - 2} }{( - 2)} + C = \frac{3}{x} - \frac{2}{ {x}^{2} } + C$

5.

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

6.

$F(x) = \int\limits( \sqrt{x} - \frac{2}{ \sqrt{x} } ) dx= \int\limits( {x}^{ \frac{1}{2} } - 2 {x}^{ - \frac{1}{2} } )dx = \\ = \frac{ {x}^{ \frac{3}{2} } }{ \frac{3}{2} } - 2 \times \frac{ {x}^{ \frac{1}{2} } }{ \frac{1}{2} } + C = \frac{2}{3}x \sqrt{x} - 4 \sqrt{x} + C$

8.

$F(x) = \int\limits( \sqrt{x} + 2 {x}^{2} \sqrt{x} )dx =\int\limits ( {x}^{ \frac{1}{2} } + 2 {x}^{ \frac{5}{2} }) dx = \\ = \frac{ {x}^{ \frac{3}{2} } }{ \frac{3}{2} } + \frac{2 {x}^{ \frac{7}{2} } }{ \frac{7}{2} } + C = \frac{2}{3}x \sqrt{x} + \frac{4}{7} {x}^{3} \sqrt{x} + C$

7.

$F(x) = \int\limits(4 {e}^{x} + {x}^{3} )dx = 4 {e}^{x} + \frac{ {x}^{4} }{4} + C$

9.

$F(x) = \int\limits (\sin(2x) + 3 \cos(3x)) dx = \\ = \frac{1}{2}\int\limits \sin(2x) d (2x) +\int\limits \cos(3x) d(3x) = \\ = - \frac{1}{2} \cos(2x) + \sin(3x) + C$

10.

$F(x) = \int\limits(4 {e}^{ - 2x} + {(x - 1)}^{3} )dx = \\ = - 2 \times ( - 2)\int\limits {e}^{ - 2x} dx + \int\limits {(x - 1)}^{3} dx = \\ = - 2\int\limits {e}^{ - 2x} d( - 2x) + \int\limits {(x - 1)}^{3} d(x - 1) = \\ = - 2 {e}^{ - 2x} + \frac{ {(x - 1)}^{4} }{4} + C$

11.

$F(x) = \int\limits( \frac{2}{ \sqrt{ x+ 3} } - \sin {}^{2} (2x) )dx = \\ = \int\limits \frac{2}{ \sqrt{x + 3} } dx - \int\limits \sin {}^{2} (2x) dx = \\ = 2\int\limits {(x + 3)}^{ - \frac{1}{2} } d(x + 3) -\int\limits \frac{1 - \cos(4x) }{2} dx = \\ = 2 \times \frac{ {(x + 3)}^{ \frac{1}{2} } }{ \frac{1}{2} } - \frac{1}{2} (\int\limits \: dx - \frac{1}{4}\int\limits \cos(4x) d(4x)) = \\ = 4 \sqrt{x + 3} - \frac{1}{2} x + \frac{1}{8} \sin(4x) + C$

12.

$F(x) = \int\limits2 \cos {}^{2} ( \frac{x}{2} ) dx = \int\limits(1 + \cos(x)) dx = \\ = x + \sin(x) + C$

13.

$F(x) = \int\limits \frac{x}{1 + x} dx = \int\limits \frac{x + 1 - 1}{ 1+ x} dx = \\ = \int\limits( \frac{x + 1}{x + 1} - \int\limits \frac{1}{x + 1} )dx = \\ = \int\limits \: dx - \int\limits \frac{d(x + 1)}{x + 1} = x - ln( |x + 1| ) + C$

14.

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