Question

Решите пожалуйста как можно быстрее​

Answer 1

Ответ:

1

$\int\limits {(x + 5)}^{7} dx = \int\limits {(x + 5)}^{7} d(x + 5) = \\ = \frac{ {(x + 5)}^{8} }{8} + C$

2

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

3

$\int\limits \sqrt[3]{ {(2 {x}^{3} + 1)}^{2} } {x}^{2} dx = \frac{1}{6} \int\limits \sqrt[3]{ {(2 {x}^{3} + 1)}^{2} } 6 {x}^{2} dx = \\ = \frac{1}{6} \int\limits \sqrt[3]{ {(2 {x}^{3} + 1)}^{2} } d(2 {x}^{3} ) = \frac{1}{6} \int\limits {(2 {x}^{3} + 1) }^{ \frac{2}{3} } d(2 {x}^{3} + 1) = \\ = \frac{1}{6} \times \frac{ {(2 {x}^{3} + 1)}^{ \frac{5}{3} } }{ \frac{5}{3} } + C= \frac{1}{10} \sqrt[3]{ {(2 {x}^{3} + 1) }^{5} } + C$

4

$\int\limits \: x {e}^{ {x}^{2} } dx = \frac{1}{2} \int\limits2x {e}^{ {x}^{2} } dx = \\ = \frac{1}{2} \int\limits {e}^{ {x}^{2} } d( {x}^{2} ) = \frac{1}{2} {e}^{ {x}^{2} } + C$

5

$\int\limits \frac{ {x}^{2} dx}{ 3{x}^{3} + 4} = \frac{1}{9} \int\limits \frac{9{x}^{2}dx }{3 {x}^ {3} + 4} = \\ = \frac{1}{9} \int\limits \frac{d(3 {x}^{3} + 4) }{3 {x}^{3} + 4 } = \frac{1}{9} ln(3 {x}^{3} + 4 ) + C$

6

$\int\limits \sin(x) \cos ^{7} (x) dx = \\ = - \int\limits { \cos}^{7} (x)d( \cos(x) ) = - \frac{ { \cos }^{8}(x) }{8} + C$