Question

Помогите с интегралами

Answer 1

Ответ:

1.

$\int\limits \frac{ \sin(x) }{ \sqrt[3]{3 + 2 \cos(x) } } dx \\ \\ 3 + 2 \cos(x) = t \\ - 2 \sin(x) dx = dt \\ \sin(x)dx = - \frac{dt}{2} \\ \\ - \frac{1}{2} \int\limits \frac{dt}{ \sqrt[3]{t} } = - \frac{1}{2} \int\limits {t}^{ - \frac{1}{3} } dt = \\ = - \frac{1}{2} \times \frac{ {t}^{ \frac{2}{3} } }{ \frac{2}{3} } + C = - \frac{3}{4} \sqrt[3]{ {t}^{2} } + C = \\ = - \frac{3}{4} \sqrt[3]{ {(3 + 2 \cos(x)) }^{2} } + C$

2.

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