Question

Вычислите интеграл
$\int\limits^{e^2}_1 {\frac{dx}{x(4+ln^2x)} } \,$

Answer 1

Ответ:

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