Question

Вычислите интегралы пожалуйста

Answer 1

Ответ:

1) 1

2) 0,5

Объяснение:

1)

$\small \int \limits_{ \frac{\pi}{4} }^{ \frac{ \pi}{2} } \frac{dx}{ { \sin}^{2}x } = ( - \text{ctg \: }x ) \bigg| ^{ \frac{\pi}{2} }_{\frac{\pi}{4}} = \\ \small = ( - \text{ctg \: } \large\tfrac{\pi}{2} \small ) - ( - \text{ctg \: } \large\tfrac{\pi}{4} \small ) = \\ \small = \text{ctg \: } \large\tfrac{\pi}{2} \small + \text{ctg \: } \large\tfrac{\pi}{4} \small = 0 + 1 = 1$

2)

$\small \int \limits_{ 1}^{2} (1 - \large \tfrac{1}{ {x}^{2} } \small )dx = \int \limits_{ 1}^{2}(1 - x^{ - 2} )dx = \\ \small = \big(x - ( - 1 \cdot{x}^{ - 2 + 1} )\big) \bigg| _{ 1}^{2} = \big(x + \large \tfrac{1}{x} \small)\big) \bigg| _{ 1}^{2} = \\ \small = \big(2 + \large \tfrac{1}{2} \small\big) - \big(1 + \large\tfrac{1}{1}\small \big) = 2\large \tfrac{1}{2} \small - 2 = \large \tfrac{1}{2} \small= 0.5$