Justify convergence of an integral

Question

How would I justify the convergence of the following integral?

$$\int_0^1 \frac{1}{1-x} + \frac{1}{\log(x)} dx$$

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So far I looked at the laurent series of $1/\log(x)$ and I tried graphing the functions involved https://www.desmos.com/calculator/jmputv2vmm to get some ideas but I don't know how to prove that two divergent terms together "cancel" to produce a convergent integral.

Answer 1

Hint: Apply L'Hopital's Rule twice to show that the integrand has limits $\frac 1 2$ at $1$. Apply L'Hopital's Rule once to show that the integrand has limits $1$ at $0$. Since it is continuous on $(0,1)$ it the integral is convergent.