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\[L(n)=\dfrac{1}{n+1}+\dfrac{1}{n+2}+ \frac 1{n+3}+...+\dfrac{1}{2n}\]

For \(L(n)\) as defined above, find ...

\[\large \int_{0}^{1}{\dfrac{\ln{(1+x)}}{1+x^{2}}dx}=\dfrac{\pi^{a}\ln{b}}{c}\]

Enter \(a+b+c\), where \(a\), \(b\) and \(c\) are positive integers.

Minimise this expression

\(\sqrt{x^{2}+(4-y)^{2}}+\sqrt{(3-x)^{2}+y^{2}}\)

Here, \(x,y\) are all real numbers.

\[\large \int_1^{\infty}{\dfrac{\ln{(1+x^{2}})}{1+x^{2}}dx}=?\]

Find your answer in closed form and give your answer to 3 decimal places.

\[\int_{-1}^1 \sin ^{-1}(x) \cos ^{-1}(x) \tan ^{-1}(x) \, dx=\frac{1}{8} \pi ^2 \left(-2 \log \left(a-b \sqrt{2}\right)-4 \sqrt{2}+\pi +4\right)\]

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