Question

Пожалуйста !!!!с подробным решением срочно!!!!

Answer 1

Ответ:

1

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

2

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

3

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

4

$\int\limits {7}^{ {x}^{3} } {x}^{2} dx \\ \\ {x}^{3} = t \\ 3 {x}^{2} dx = dt \\ {x}^{2} dx = \frac{dt}{3} \\ \\ \frac{1}{3} \int\limits {7}^{t} dt = \frac{1}{3} \times \frac{ {7}^{t} }{ ln(7) } + C= \frac{ {7}^{ {x}^{3} } }{3 ln(7) } + C$

5

$\int\limits \frac{ \sin(x) }{ 1 - \cos(x) } dx \\ \\ 1 - \cos(x) = t \\ \sin(x) dx = dt \\ \\ \int\limits \frac{dt}{t} = ln |t| + C= ln |1 - \cos(x) | + C$