-2*pi*(log(sin(pi*x)+1)*1/2-log(sin(pi*x)-1)*1/2) если x=-1/3 (упростите выражение)

Выражение, которое надо упростить:

Решение

Вы ввели
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      /log(sin(pi*x) + 1)   log(sin(pi*x) - 1)\
-2*pi*|------------------ - ------------------|
      \        2                    2         /
$$- 2 \pi \left(- \frac{1}{2} \log{\left (\sin{\left (\pi x \right )} - 1 \right )} + \frac{1}{2} \log{\left (\sin{\left (\pi x \right )} + 1 \right )}\right)$$
Подстановка условия
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(-2*pi)*(log(sin(pi*x) + 1)/2 - log(sin(pi*x) - 1)/2) при x = -1/3
(-2*pi)*(log(sin(pi*x) + 1)/2 - log(sin(pi*x) - 1)/2)
$$- 2 \pi \left(- \frac{1}{2} \log{\left (\sin{\left (\pi x \right )} - 1 \right )} + \frac{1}{2} \log{\left (\sin{\left (\pi x \right )} + 1 \right )}\right)$$
(-2*pi)*(log(sin(pi*(-1/3)) + 1)/2 - log(sin(pi*(-1/3)) - 1)/2)
$$- 2 \pi \left(- \frac{1}{2} \log{\left (\sin{\left (\pi (-1/3) \right )} - 1 \right )} + \frac{1}{2} \log{\left (\sin{\left (\pi (-1/3) \right )} + 1 \right )}\right)$$
(-2*pi)*(log(sin(pi*(-1)/3) + 1)/2 - log(sin(pi*(-1)/3) - 1)/2)
$$- 2 \pi \left(\frac{1}{2} \log{\left (\sin{\left (\frac{-1 \pi}{3} \right )} + 1 \right )} - \frac{1}{2} \log{\left (-1 + \sin{\left (\frac{-1 \pi}{3} \right )} \right )}\right)$$
-2*pi*(log(1 - sqrt(3)/2)/2 - log(1 + sqrt(3)/2)/2 - pi*i/2)
$$- 2 \pi \left(\frac{1}{2} \log{\left (- \frac{\sqrt{3}}{2} + 1 \right )} - \frac{1}{2} \log{\left (\frac{\sqrt{3}}{2} + 1 \right )} - \frac{i \pi}{2}\right)$$
Численный ответ
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3.14159265358979*log(sin(pi*x) - 1) - 3.14159265358979*log(sin(pi*x) + 1)
Рациональный знаменатель
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pi*(-log(1 + sin(pi*x)) + log(-1 + sin(pi*x)))
$$\pi \left(\log{\left (\sin{\left (\pi x \right )} - 1 \right )} - \log{\left (\sin{\left (\pi x \right )} + 1 \right )}\right)$$
Объединение рациональных выражений
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-pi*(-log(-1 + sin(pi*x)) + log(1 + sin(pi*x)))
$$- \pi \left(- \log{\left (\sin{\left (\pi x \right )} - 1 \right )} + \log{\left (\sin{\left (\pi x \right )} + 1 \right )}\right)$$
Общее упрощение
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pi*(-log(1 + sin(pi*x)) + log(-1 + sin(pi*x)))
$$\pi \left(\log{\left (\sin{\left (\pi x \right )} - 1 \right )} - \log{\left (\sin{\left (\pi x \right )} + 1 \right )}\right)$$
Собрать выражение
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pi*log(-1 + sin(pi*x)) - pi*log(1 + sin(pi*x))
$$\pi \log{\left (\sin{\left (\pi x \right )} - 1 \right )} - \pi \log{\left (\sin{\left (\pi x \right )} + 1 \right )}$$
      /-1 + sin(pi*x)\
pi*log|--------------|
      \1 + sin(pi*x) /
$$\pi \log{\left (\frac{\sin{\left (\pi x \right )} - 1}{\sin{\left (\pi x \right )} + 1} \right )}$$
Общий знаменатель
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pi*log(-1 + sin(pi*x)) - pi*log(1 + sin(pi*x))
$$\pi \log{\left (\sin{\left (\pi x \right )} - 1 \right )} - \pi \log{\left (\sin{\left (\pi x \right )} + 1 \right )}$$
Тригонометрическая часть
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-pi*(-log(-1 + sin(pi*x)) + log(1 + sin(pi*x)))
$$- \pi \left(- \log{\left (\sin{\left (\pi x \right )} - 1 \right )} + \log{\left (\sin{\left (\pi x \right )} + 1 \right )}\right)$$
Комбинаторика
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-pi*(-log(-1 + sin(pi*x)) + log(1 + sin(pi*x)))
$$- \pi \left(- \log{\left (\sin{\left (\pi x \right )} - 1 \right )} + \log{\left (\sin{\left (\pi x \right )} + 1 \right )}\right)$$