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∫ Ci ( a x ) Ci ( b x ) d x = ( x Ci ( a x ) − sin a x a ) Ci ( b x ) − ∫ ( x Ci ( a x ) − sin a x a ) cos b x x d x {\displaystyle \int {\text{Ci}}(ax)\,{\text{Ci}}(bx)\,dx=\left(x\,{\text{Ci}}(ax)-{\frac {\sin ax}{a}}\right){\text{Ci}}(bx)-\int \left(x\,{\text{Ci}}(ax)-{\frac {\sin ax}{a}}\right){\frac {\cos bx}{x}}\,dx} = x Ci ( a x ) Ci ( b x ) − sin a x a Ci ( b x ) − ∫ Ci ( a x ) cos b x d x + ∫ sin a x a cos b x x d x {\displaystyle =x\,{\text{Ci}}(ax)\,{\text{Ci}}(bx)-{\frac {\sin ax}{a}}\,{\text{Ci}}(bx)-\int {\text{Ci}}(ax)\,\cos bx\,dx+\int {\frac {\sin ax}{a}}\,{\frac {\cos bx}{x}}\,dx} dabei ist ∫ sin a x a cos b x x d x = 1 2 a ∫ sin ( a x + b x ) + sin ( a x − b x ) x d x = 1 2 a ( Si ( a x + b x ) + Si ( a x − b x ) ) {\displaystyle \int {\frac {\sin ax}{a}}\,{\frac {\cos bx}{x}}\,dx={\frac {1}{2a}}\int {\frac {\sin(ax+bx)+\sin(ax-bx)}{x}}\,dx={\frac {1}{2a}}{\Big (}{\text{Si}}(ax+bx)+{\text{Si}}(ax-bx){\Big )}} und ∫ Ci ( a x ) cos b x d x = Ci ( a x ) sin b x b − ∫ cos a x x sin b x b d x {\displaystyle \int {\text{Ci}}(ax)\,\cos bx\,dx={\text{Ci}}(ax)\,{\frac {\sin bx}{b}}-\int {\frac {\cos ax}{x}}\,{\frac {\sin bx}{b}}\,dx} , wobei hier wiederum ∫ cos a x x sin b x b d x = 1 2 b ∫ sin ( a x + b x ) − sin ( a x − b x ) x d x = 1 2 b ( Si ( a x + b x ) − Si ( a x − b x ) ) {\displaystyle \int {\frac {\cos ax}{x}}\,{\frac {\sin bx}{b}}\,dx={\frac {1}{2b}}\int {\frac {\sin(ax+bx)-\sin(ax-bx)}{x}}\,dx={\frac {1}{2b}}{\Big (}{\text{Si}}(ax+bx)-{\text{Si}}(ax-bx){\Big )}} ist.
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