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$$\int_0^{+\infty}e^{-t^2}dt\:=\:\frac{\sqrt{\pi}}{2}$$ |
$$\int_0^{\frac{\pi}{2}}\frac{\sin(x)}{x}dx\:=\:\frac{\pi}{2}$$ |
$$\int_0^{+\infty}\frac{\sin^2(t)}{t^2}dt\:=\:\frac{\pi}{2}$$ |
$$\int_0^{+\infty}\frac{\sin^3(t)}{t^2}dt\:=\:\frac{3\ln(3)}{4}$$ |
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$$\int_0^{\frac{\pi}{2}}\ln(\sin(t))dt\:=\:\int_0^{\frac{\pi}{2}}\ln(\cos(t))dt\:=\:$$ |
$$\int_0^1\ln^n(t)dt$$ |
$$\int_0^{+\infty}\frac{1}{(1+t^2)^n}dt$$ |
$$\int_0^1\frac{x-1}{\ln(x)}dx\:=\:\ln(2)$$ |
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$$\int_0^{\frac{\pi}{2}}sin^{2n}(x)dx\:=\:\frac{(2n)!}{2^{2n}(n!)^2}\frac{\pi}{2}$$ |
$$\int_0^{\frac{\pi}{2}}sin^{2n+1}(x)dx\:=\:\frac{2^{2n}(n!)^2}{(2n+1)!}$$ |
$$\int_0^{+\infty}\frac{1}{1+t^4}dt$$ |
$$\int_0^{+\infty}\frac{e^{-x}-e^{-2x}}{x}dx$$ |
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$$\int_0^{+\infty}\frac{\cos(at)-\cos(bt)}{t}dt$$ |
$$\int_0^{+\infty}\!\!\frac{t\ln(t)}{(1+t^2)^2}dt$$ |
$$\int_0^{\frac{\pi}{2}}\frac{t}{\tan(t)}dt\:=\:\frac{\pi}{2}\ln(2)$$ |
$$\int_0^{\frac{\pi}{2}}\ln(\sin(t))dt\:=\:-\frac{\pi}{2}\ln(2)$$ |
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$$\int_0^{+\infty}\frac{t}{e^t-1}dt\:=\:\frac{\pi^2}{6}$$ |
$$\\int_0^{+\infty}e^{-t}\:\frac{\sin(xt)}{t}dt$$ |
$$\int_0^{+\infty}e^{-t^2}\cos(xt)dt$$ |
$$\int_0^{1}\frac{\ln(1-t)\ln(t)}{t}dt\:=\:\zeta(3)$$ |
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$$\int_0^{+\infty}\frac{1-e^{-xt^2}}{t^2}dt$$ |
$$\int_0^1\frac{\arctan(x)}{x}dx\:=\:\sum_{k=0}^{+\infty}\frac{(-1)^{k}}{(2k+1)^2}$$ |
$$\int_0^1\frac{\ln(t)}{1-t^2}dt\:=\:-\frac{\pi^2}{8}$$ | |
