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Math Help - integral of 1/x

  1. #1
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    integral of 1/x

    Hi,

    I understand that differentiate both ln x and ln |x| will give 1/x.
    But why \int \frac{1}{x} dx=ln|x|+c.
    Is it ok if I write \int \frac{1}{x} dx=ln x+c

    Can someone help?
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by acc100jt View Post
    Hi,

    I understand that differentiate both ln x and ln |x| will give 1/x.
    But why \int \frac{1}{x} dx=ln|x|+c.
    Is it ok if I write \int \frac{1}{x} dx=ln x+c

    Can someone help?
    It depends on the interval x is defined on.

    If x\in\left(0,\infty\right), then its safe to say that \int\frac{1}{x}\,dx=\ln x+C. However, if x\in\left(-\infty,0\right), then \int\frac{1}{x}\,dx=\ln\left(-x\right)+C.

    Thus, it would be best to define \int\frac{1}{x}\,dx=\ln\!\left|x\right|+C for x\in\mathbb{R}\backslash\{0\} (any real number except zero)
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