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Triple integral with sines and cosines

Find the integral $$\begin{align}\int^\infty_0 \int^\infty_0 \int^\infty_0 \frac{\sin(x)\sin(y)\sin(z)}{xyz(x+y+z)}(\sin(x)\cos(y)\cos(z)\\ + \sin(y)\cos(z)\cos(x) + \sin(z)\cos(x)\cos(y))\,dx\,dy\,dz \end{align}$$   $$\textit{solution}$$ This can be rewritten as $$3\small\int^\infty_0 \int^\infty_0 \int^\infty_0 \frac{\sin^2(x)\sin(y)\cos(y)\sin(z)\cos(z)}{xyz(x+y+z)}\,dx\,dy\,dz$$ Now consider $$\small F(a) = 3\int^\infty_0 \int^\infty_0 \int^\infty_0\frac{\sin^2(x)\sin(y)\cos(y)\sin(z)\cos(z) e^{-a(x+y+z)}}{xyz(x+y+z)}\,dx\,dy\,dz$$ Taking the derivative $$\small F'(a) = -3\int^\infty_0 \int^\infty_0 … Continue reading

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Cosine sine integral

Prove that $$\int_0^{2\pi} \cos(\sin{x})e^{\cos{x}} dx = 2\pi$$ $$\textit{proof}$$ $$\begin{align} \int_0^{2\pi} \cos(\sin{x})e^{\cos{x}} dx &=Re\left( \int_0^{2\pi} e^{i\sin(x)}e^{\cos{x}} dx\right) \\ &= Re \left(\int_0^{2\pi} e^{e^{i x}} dx\right)\\ &= Re \left(\sum_{n=0}^\infty \frac{1}{n!}\int_0^{2\pi} e^{inx} dx\right) \\ &= \sum_{n=0}^\infty \frac{1}{n!}\int_0^{2\pi} \cos(nx) dx \\ &= \int_0^{2\pi} dx = … Continue reading

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Relation between binomial sum and harmonic numbers

Prove that $$\sum_{r=1}^n {n\choose r}(-1)^{r+1}\dfrac{1}{r}=\sum_{r=1}^n \dfrac{1}{r}$$ $$proof$$ Start by $$\sum_{r=0}^n {n\choose r}x^r=(1+x)^n$$ Which can be converted to integration $$\sum_{r=1}^n {n\choose r}\frac{(-1)^{r}}{r}=\int^{-1}_0 \frac{(x+1)^n-1}{x} dx$$ By substitution we have $$\sum_{r=1}^n {n\choose r}\frac{(-1)^{r+1}}{r}=\int^{1}_0 \frac{t^n-1}{t-1} dt = H_n$$ Note the last step by expanding … Continue reading

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