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Find the integral of 1/(x log x)

Evaluate the following integral:

    \[ \int \frac{dx}{x \log x}. \]


First, we add and subtract \log x in the numerator,

    \begin{align*}   \int \frac{1}{x \log x} \, dx &= \int \frac{1+\log x - \log x}{x \log x} \, dx \\  &= \int \frac{1+\log x}{x \log x} \, dx - \int \frac{\log x}{x \log x} \, dx \\  &= \int \frac{1+\log x}{x \log x} \, dx - \int \frac{1}{x} \, dx \\  &= \int \frac{1+\log x}{x \log x} \, dx - \log |x| + C. \end{align*}

For the integral on the left, let u = x \log x then du = 1 + \log x and so we have

    \begin{align*}    \int \frac{1+\log x}{x \log x} \, dx &= \int \frac{du}{u} \\  &= \log |u| + C \\  &= \log |x \log x| + C\\  &= \log |x| + \log | \log x| + C. \end{align*}

Therefore, we have

    \begin{align*}  \int \frac{dx}{x \log x} &= \int \frac{1+\log x}{x \log x} \, dx - \log |x| + C \\  &= \log |x| + \log |\log x| - \log|x| + C \\  &= \log|\log x| + C. \end{align*}

2 comments

  1. Anonymous says:

    what if you just let u = log x du = 1/x dx so then you have the integral of 1/u du, integrate and you get log u + c. subsititute to obtain: log | log x| + c

  2. chuck says:

    need help to find the integral ln(x)/(x+a). Really appreciate it if you can show the integral can be expressed in elementary functions.

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