Question

найти первообразную функций:

Answer 1

11.

$F(x) = \int\limits \frac{dx}{10 - 5x} = - \frac{1}{5} \int\limits \frac{d(10 - 5x)}{1 0 - 5x} = \\ = - \frac{1}{5} ln(10 - 5x) + c$

12.

$F(x) = \int\limits {( \frac{x + 3}{2}) }^{2} dx = \int\limits \frac{ {(x + 3)}^{2} }{4} dx = \\ = \frac{1}{4} \int\limits {(x + 3)}^{2} d(x + 3) = \frac{1}{4} \times \frac{ {(x + 3)}^{3} }{3} + c = \\ = \frac{ {(x + 3)}^{3} }{12} + c$

13.

$F(x) = \int\limits( {2}^{x + 3} \times {5}^{x+ 2} )dx = \int\limits2 \times {2}^{2 + x} \times {5}^{2 + x} dx = \\ = 2\int\limits {10}^{2 + x} d(2 + x) = \frac{2}{\ln(10)} \times {10}^{2 + x} + c$

14.

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