int((d)/(dx)(1)/(1+log x)int1dx)dx

Integrate the derivative: Now we integrate the derivative we found with respect to x. The integral of -1/(x(1 + log(x))^2) with respect to x is what we are looking for. This is not a standard integral, and it does not simplify easily to a basic form. We may need to use integration techniques such as integration by parts or substitution, but in this case, these techniques do not seem to simplify the integral. Since the integral of a derivative of a function is the function itself (up to a constant), we can use the Fundamental Theorem of Calculus to state that the integral of the derivative of 1/(1+log(x)) with respect to x is simply 1/(1+log(x)) plus a constant of integration.

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Calculus

Find indefinite integrals using the substitution

Full solution

\int\left(\frac{d}{dx}\left(\frac{1}{1+\log x}\right)\int 1 dx\right)dx

Integrate the derivative: Now we integrate the derivative we found with respect to $x$ . The integral of $-\frac{1}{x(1 + \log(x))^2}$ with respect to $x$ is what we are looking for. This is not a standard integral, and it does not simplify easily to a basic form. We may need to use integration techniques such as integration by parts or substitution, but in this case, these techniques do not seem to simplify the integral. Since the integral of a derivative of a function is the function itself (up to a constant), we can use the Fundamental Theorem of Calculus to state that the integral of the derivative of $\frac{1}{1+\log(x)}$ with respect to $x$ is simply $\frac{1}{1+\log(x)}$ plus a constant of integration.