Question

Неопределенный интеграл.

Answer 1

Ответ:

1.

$\int\limits \frac{ \sin(3t) }{ \sqrt{16 + \cos {}^{2} (3t) } } dt = \frac{1}{3} \int\limits \frac{ \sin(3t) d(3x)}{ \sqrt{1 6 + \cos {}^{2} (3t) } } = \\ = - \frac{1}{3} \int\limits \frac{ - \sin(3t))d(3t) }{ \sqrt{16 + \cos {}^{2} (3t) } } = - \frac{1}{3} \int\limits \frac{d (\cos(3t)) }{ \sqrt{ {4}^{2} + \cos {}^{2} (3t) } } = \\ = - \frac{1}{3} ln( | \cos(3t) + \sqrt{16 + \cos { }^{2} (3t) } | ) + C$

2.

$\int\limits \frac{5 {arctg}^{3} {}^{} x}{1 + {x}^{2} } dx = \int\limits \frac{1}{1 + {x}^{2} } \times 5arctg {}^{3} xdx = \\ = \int\limits5arctg {}^{3} x \times d(arctgx) = \frac{5arctg {}^{4} x}{4} + C$