-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:
Например, 1/(a*x-1)-1/(a*x+1)

    Решение

    Вы ввели
    [LaTeX]
          /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
    [LaTeX]
    (-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
    [LaTeX]
    3.14159265358979*log(sin(pi*x) - 1) - 3.14159265358979*log(sin(pi*x) + 1)
    Рациональный знаменатель
    [LaTeX]
    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
    [LaTeX]
    -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
    [LaTeX]
    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
    [LaTeX]
    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
    [LaTeX]
    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
    [LaTeX]
    -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
    [LaTeX]
    -pi*(-log(-1 + sin(pi*x)) + log(1 + sin(pi*x)))
    $$- \pi \left(- \log{\left (\sin{\left (\pi x \right )} - 1 \right )} + \log{\left (\sin{\left (\pi x \right )} + 1 \right )}\right)$$