Question

Знайти визначений інтеграл

Answer 1

Ответ:

$\int\limits^{ \frac{\pi}{2} } _ {0} \frac{dx}{4 {x}^{2} + 16 } = \frac{1}{4} \int\limits^{ \frac{\pi}{2} } _ {0} \frac{dx}{ {x}^{2} + 16 } = \\ = \frac{1}{4} \int\limits^{ \frac{ \pi}{2} } _ {0} \frac{d(2x)}{ {x}^{2} + {2}^{2} } = \frac{1}{4 \times 2} arctg( \frac{x}{2} ) | ^{ \frac{\pi}{2} } _ {0} = \\ = \frac{1}{8}( arctg( \frac{\pi}{4} ) - arctg(0)) = \\ = \frac{1}{8} \times 1 = \frac{1}{8}$