Question

Первообразованная! Решите 1 вариант!!! С поиснениями за 10 минут!!

Answer 1

1 вариант

$1)F(x) = \int\limits(7 {x}^{5} + 2)dx = \\ = \frac{7 {x}^{6} }{6} + 2x + C$

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

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

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

$5)f(x) = \int\limits \frac{7}{ {x}^{2} } dx = 7\int\limits {x}^{ - 2} dx = \\ = 7 \times \frac{ {x}^{ - 1} }{( - 1)} + C = - \frac{7}{x} + C$

$6)F(x) = \int\limits \sin(3x + 4) dx = \\ = \frac{1}{3} \int\limits \sin(3x + 4) d(3x + 4) = \\ = - \frac{1}{3} \cos(3x + 4) + C$

$7)F(x) = \int\limits( \frac{4}{x} + {e}^{2x} - \cos(x) )dx = \\ = 4 ln(x) + \frac{1}{2} \int\limits {e}^{2x} d(2x) - \sin(x) + C = \\ = 4 ln(x) + \frac{1}{2} {e}^{2x} - \sin(x) + C$

2 вариант

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

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

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

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

$5)F(x) = \int\limits8 {x}^{ - 2} dx = \\ = 8 \times \frac{ {x}^{ - 1} }{( - 1)} + C = - \frac{8}{x} + C$

$6)F(x) = \int\limits \cos(3x + 4) = \\ = \frac{1}{3} \int\limits \cos(3x + 4) d(3x + 4) = \\ = \frac{1}{3} \sin(3x + 4) + C$

$7)F(x) = \int\limits (\frac{2}{x} + {e}^{x} - \sin(x) )dx = \\ = 2 ln(x) + {e}^{x} + \cos(x) + C$