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Thread: 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 $\displaystyle \int \frac{1}{x} dx=ln|x|+c$.
    Is it ok if I write $\displaystyle \int \frac{1}{x} dx=ln x+c$

    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 $\displaystyle \int \frac{1}{x} dx=ln|x|+c$.
    Is it ok if I write $\displaystyle \int \frac{1}{x} dx=ln x+c$

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

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

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