Formelsammlung Mathematik: Bestimmte Integrale: Form R(x,exp,sin)

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1.1

$\underset{0}{\overset{\mathrm{\infty }}{\int }}{\left(\frac{\sin x}{x}\right)}^2{e}^{-2ax}dx=a\log \left(\frac{a}{\sqrt{1+a^2}}\right)+\mathrm{arccot}a$

2.1

$\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-\alpha x}{\sin }^{2n}xdx=\frac{(2n)!}{\alpha ({\alpha }^2+2^2)({\alpha }^2+4^2)\cdots ({\alpha }^2+(2n{)}^2)}n\in \mathbb{N},\text{Re}(\alpha )>0$

2.2

$\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-\alpha x}{\sin }^{2n+1}xdx=\frac{(2n+1)!}{({\alpha }^2+1)({\alpha }^2+3^2)\cdots ({\alpha }^2+(2n+1{)}^2)}n\in \mathbb{N},\text{Re}(\alpha )>0$

2.3

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\sin \alpha x}{1-e^{\beta x}}dx=\frac{1}{2\alpha }-\frac{\pi }{2\beta }\text{coth}\left(\frac{\alpha \pi }{\beta }\right)$

2.4

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\sin \alpha x}{1+e^{\beta x}}dx=\frac{1}{2\alpha }-\frac{\pi }{2\beta }\text{csch}\left(\frac{\alpha \pi }{\beta }\right)$

2.5

$\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-ax}\sin bxdx=\frac{b}{a^2+b^2}$

2.6

$\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-ax}\frac{\sin bx}{x}dx=\arctan \left(\frac{b}{a}\right)\text{Re}(a)\ge |\text{Im}(b)|,\frac{b}{a}\ne \pm i$

3.1

$\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-ax}\sin (bx)x^{s-1}dx=\frac{\mathrm{\Gamma }(s)}{{\sqrt{a^2+b^2}}^s}\sin (s\arctan \frac{b}{a})a>0,b\in ℝ,\text{Re}(s)>0$