Prove that [latex]\int_{0}^{1}{\left(\frac{\log x}{1 − x}\right)^{2}} dx = 2 \sum\limits_{n=1}^{∞}{\frac{1}{n^{2}}}.[/latex]

For every x ∈ (−1, 1) we have the power series representation [latex]\frac{1}{1 − x} = \sum\limits_{n=0}^{∞}{x^{n}}.[/latex] Differentiating both sides, we obtain [latex]\frac{1}{(1 − x)^{2}} = \sum\limits_{n=1}^{∞}{nx^{n−1}}.[/latex] Using Corollary 11.3.1, we can write [latex]\int_{0}^{1}{\left(\frac{\log x}{1 − x}\right)^{2}} dx = \sum\limits_{n=1}^{∞}{n \int_{0}^{1}{x^{n−1}(\log x)^{2}} dx}.[/latex] (11.30) When n = 1 the integral [latex]\int_{0}^{1} (\log x)^{2} dx[/latex] on the right-hand side is improper, whereas the other integrals are Riemann integrals. In either case we use integration by parts (and L’Hôpital’s rule) to obtain [latex]\int_{0}^{1}{x^{n−1}(\log x)^{2}} dx = \frac{x^{n}}{n} (\log x)^{2} \bigg|_{0}^{1} − \int_{0}^{1}{\frac{2x^{n−1} \log x}{n}} dx[/latex] [latex]= − \frac{2}{n} \left(\frac{x^{n} \log x}{n} \bigg|_{0}^{1} − \int_{0}^{1}{\frac{x^{n−1}}{n}} dx \right)[/latex] [latex]= \frac{2}{n^{3}}.[/latex] Substituting into (11.30) yields the desired equality.

Question 11.10: Prove that ∫¹0 (log x/1 − x)² dx = 2 ∑n=1^∞ 1/n².

Elements of Real Analysis

Prove that

$\int_{0}^{1}{\left(\frac{\log x}{1 − x}\right)^{2}} dx = 2 \sum\limits_{n=1}^{∞}{\frac{1}{n^{2}}}.$

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For every x ∈ (−1, 1) we have the power series representation

$\frac{1}{1 − x} = \sum\limits_{n=0}^{∞}{x^{n}}.$

Differentiating both sides, we obtain

$\frac{1}{(1 − x)^{2}} = \sum\limits_{n=1}^{∞}{nx^{n−1}}.$

Using Corollary 11.3.1, we can write

$\int_{0}^{1}{\left(\frac{\log x}{1 − x}\right)^{2}} dx = \sum\limits_{n=1}^{∞}{n \int_{0}^{1}{x^{n−1}(\log x)^{2}} dx}.$ (11.30)

When n = 1 the integral $\int_{0}^{1} (\log x)^{2} dx$ on the right-hand side is improper, whereas the other integrals are Riemann integrals. In either case we use integration by parts (and L’Hôpital’s rule) to obtain

$\int_{0}^{1}{x^{n−1}(\log x)^{2}} dx = \frac{x^{n}}{n} (\log x)^{2} \bigg|_{0}^{1} − \int_{0}^{1}{\frac{2x^{n−1} \log x}{n}} dx$

$= − \frac{2}{n} \left(\frac{x^{n} \log x}{n} \bigg|_{0}^{1} − \int_{0}^{1}{\frac{x^{n−1}}{n}} dx \right)$

$= \frac{2}{n^{3}}.$

Substituting into (11.30) yields the desired equality.