Question

Помогите пожалуйста ​

Answer 1

Ответ:

13.

$\int\limits \: x(5x - 7) {}^{50} dx = \frac{1}{5} \int\limits5x {(5x - 7)}^{50} dx = \\ = \frac{1}{5} \int\limits(5x - 7 + 7) {(5x - 7)}^{50} dx = \\ = \frac{1}{5} \int\limits(5x - 7) {(5x - 7)}^{50} + 7 {(5x - 7)}^{50} )dx = \\ = \frac{1}{5} \int\limits {(5x - 7)}^{51} dx + \frac{1}{5} \times 7\int\limits {(5x - 7)}^{50} dx = \\ = \frac{1}{25} \int\limits {(5x - 7)}^{51} d(5x - 7) + \frac{7}{25} \int\limits {(5x - 7)}^{50} d(5x - 7) = \\ = \frac{ {(5x - 7)}^{52} }{25 \times 52} + \frac{7}{25 \times 51} {(5x - 7)}^{51} + C = \\ = \frac{1}{25} ( \frac{ {(5x - 7)}^{52} }{52} + \frac{7 {(5x - 7)}^{51} }{51} ) + C$

14.

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

15.

$\int\limits \frac{5x - 6}{ \sqrt{1 - 3x} } dx = 5\int\limits \frac{x - \frac{6}{5} }{ \sqrt{1 - 3x} } dx = \\ = 5 \times ( - \frac{1}{3} )\int\limits \frac{( - 3x + \frac{18}{5}) }{ \sqrt{1 - 3x} } dx = \\ = - \frac{5}{3} \int\limits \frac{ (- 3x + 1 + \frac{13}{5} }{ \sqrt{1 - 3x} } dx = \\ = - \frac{5}{3} \int\limits \frac{1 -3 x}{ \sqrt{1 - 3x} } - \frac{5}{3} \times \frac{13}{5} \int\limits \frac{dx}{ \sqrt{1 - 3x} } = \\ = - \frac{5}{3} \int\limits \sqrt{1 - 3x} dx - \frac{13}{3} \times ( - \frac{1}{3}) \int\limits \frac{d(1 - 3x)}{ \sqrt{1 - 3x} } = \\ = - \frac{5}{3} \times ( - \frac{1}{3} )\int\limits {(1 - 3x)}^{ \frac{1}{2} } d(1 - 3x) + \frac{13}{9} \times \frac{ {(1 - 3x)}^{ \frac{1}{2} } }{ \frac{1}{2} } + C = \\ = \frac{5}{9} \times \frac{ {(1 - 3x)}^{ \frac{3}{2} } }{ \frac{3}{2} } + \frac{26}{9} \sqrt{1 - 3x} + c = \\ = \frac{10}{27} \sqrt{ {(1 - 3x)}^{3} } + \frac{26}{9} \sqrt{1 - 3x} + C$

16.

$\int\limits {x}^{2} \sqrt[5]{ {x}^{3} - 8 }d x \\ \\ {x}^{3} - 8 = t \\ 3 {x}^{2} dx = dt \\ {x}^{2} dx = \frac{dt}{3} \\ \\ \frac{1}{3} \int\limits \sqrt[5]{t} dt = \frac{1}{3} \int\limits {t}^{ \frac{1}{5} } dt = \frac{1}{3} \times \frac{ {t}^{ \frac{6}{5} } }{ \frac{6}{5} } + C = \\ = \frac{5}{18} \sqrt[5]{ {x}^{3} - 8} + C$