Equipped with the right applications, a computer can be of great help in virtually any domain of activity. When it comes to designing and precision, no other tool is as accurate as a computer. Moreover, specialized applications such as AutoCAD give you the possibility to design nearly anything ranging from art, to complex mechanical parts or even buildings.
Suitable for business environments and experienced users
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To end with
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Q:

Evaluate the integral $\int_0^1{\ln (1+x)\ln (1-x)\,dx}$

$$\int_0^1{\ln (1+x)\ln (1-x)\,dx}$$
I think that this could be solved by using Wolfram alpha.
But how should I do it by hand?
Any help will be appreciated.

A:

Note that the function is continuous and we have
$$\int_0^1{\ln(1+x)\ln(1-x)\,dx}=\int_{ -1}^1{\ln(1+x)\ln(1-x)\,dx}= \int_{ -1}^1{\ln^2(1+x)\,dx}$$
since the integrand is even.
Let $u=1+x$. By the change of variables $x\mapsto1+x$ we have
$$\int_{ -1}^1{\ln^2(1+x)\,dx}=\frac12\int_{ -1}^1{\ln^2(1+2x)\,dx}= \frac12\int_{ -1}^1{\ln^2(u)\,du}=\frac12\int_0^2{\ln^2(u)\,du}=\frac12\left[\ln^3(2)\right]_0^2-\frac14\ln^4(2)$$
hence
$$\int_0^1{\ln(1+x)\ln(1-x)\,dx}=\frac18\ln^4(2)$$

A:

Applying the double change of variables $x \to 1-x$ and \$

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