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Zacky
I solve integrals for fun.
Here are some that I really enjoyed solving (in no particular order):
- Why does $\int_1^\sqrt2 \frac{1}{x}\ln\left(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\right)dx$ equal to $0$?
- Evaluating $\int_0^{\infty} \frac{1}{1-x^2+x^4}\ln\left(\frac{x^2}{1-x^2+x^4}\right)dx$
- Evaluate $\int_0^{\infty} \frac {\ln(1+x^3)}{1+x^2}dx$
- Quotient of two integrals $\frac{\int_0^\pi x^3\ln(\sin x)dx}{\int_0^\pi x^2\ln(\sqrt{2}(\sin x)dx}$
- Baaad: $\int_0^{\pi/2}\log\bigg(\frac{2+\sin2x}{2-\sin2x}\bigg)\mathrm dx$
- Integral $\int_{0}^{1} \int_{0}^{1} \frac{1}{(1+x y) \ln (x y)} d x d y$
- Seeking Methods to solve $\int_{0}^{\frac{\pi}{2}} \ln\left|\sec^2(x) + \tan^4(x) \right|\:dx $
- Evaluating $\int_0^1\frac{\ln(1+x-x^2)}xdx$ without using polylogarithms.
- Evaluating $\int_1^2\frac{\arctan(x+1)}{x}\,dx$
- Closed form of $\int_0^\pi \ln\left(1+\sin^2(t)\right) dt$?
- Show that $\sum\limits_{n=1}^{\infty}\frac1{n^2}=\sum\limits_{n=1}^{\infty}\frac3{n^2\binom{2n}n}$ without actually evaluating both series
- Integral $\small \int_0^\infty \frac{(\pi x - 2\log{x})^3}{\left(\log^2{x} + \frac{\pi^2}{4}\right)(1+x^2)^2} \text{d}x$
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