Compute $\int\limits_0^1 \log (2\arctan x+\frac{2x}{1+x^2})(1+x\arctan x)\ dx$

Question

How to Find the value of $$I=\int_{0}^{1}\log \Big(2\arctan x+\frac{2x}{1+x^2}\Big)\Big(1+x\arctan x\Big)dx$$

I tried to subsititution:x=tant, but the process became very complicated,I don't know how to deal with it, and any help will be appreciated.

Answer 1

Well, we have:

$$\mathcal{I}:=\int\limits_0^1\ln\left(2\cdot\arctan\left(x\right)+\frac{2x}{1+x^2}\right)\cdot\left(1+x\cdot\arctan\left(x\right)\right)\space\text{d}x\tag1$$

Using integration by parts:

$$\mathcal{I}=\frac{2+\pi}{4}\cdot\ln\left(\frac{2+\pi}{2}\right)-\int\limits_0^1\frac{1}{1+x^2}\space\text{d}x=\frac{2+\pi}{4}\cdot\ln\left(\frac{2+\pi}{2}\right)-\frac{\pi}{4}\tag2$$

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Integration by parts is used as follows:

$$\int\text{f}\left(x\right)\cdot\text{g}\space'\left(x\right)\space\text{d}x=\text{f}\left(x\right)\cdot\text{g}\left(x\right)-\int\text{f}\space'\left(x\right)\cdot\text{g}\left(x\right)\space\text{d}x\tag3$$

Where:

• $$\text{f}\left(x\right):=\ln\left(2\cdot\arctan\left(x\right)+\frac{2x}{1+x^2}\right)\tag4$$
• $$\text{g}\space'\left(x\right):=1+x\cdot\arctan\left(x\right)\tag5$$

And then you'll find:

• \begin{equation} \begin{split} \left[\text{f}\left(x\right)\cdot\text{g}\left(x\right)\right]_0^1&=\lim_{x\space\to\space1}\left(\text{f}\left(x\right)\cdot\text{g}\left(x\right)-\text{f}\left(x-1\right)\cdot\text{g}\left(x-1\right)\right)\\ \\ &=\text{f}\left(1\right)\cdot\text{g}\left(1\right)-\lim_{x\space\to\space0}\text{f}\left(x\right)\cdot\text{g}\left(x\right)\\ \\ &=\frac{2+\pi}{4}\cdot\ln\left(\frac{2+\pi}{2}\right) \end{split}\tag6 \end{equation}

• $$\text{f}\space'\left(x\right)\cdot\text{g}\left(x\right)=\frac{1}{1+x^2}\tag7$$