Question

Интегралы СРОЧНО 40 балов

Answer 1

Ответ:

$\int\limits \sqrt[4]{ \cos {}^{3} (x) } \sin(x) dx \\ \\ \cos(x) = t \\ - \sin(x) dx = dt \\ \\ - \int\limits \sqrt[4]{ {t}^{3} } dt = -\int\limits t {}^{ \frac{3}{4} }d t = - \frac{ {t}^{ \frac{7}{4} } }{ \frac{7}{4} } + C = \\ = - \frac{4}{7} \sqrt[4]{ {t}^{7} } + C = - \frac{4}{7} \sqrt[4]{ \cos {}^{7} (x) } + C$

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