Dodano po 1 dniu 2 godzinach 56 minutach 59 sekundach:
Nikt nie chce się pobawić
Ukryta treść:
Całka nie jest trudna do policzenia o ile nie podążamy za tą amerykańską modą i nie unikamy całkowania przez części
Korzystając z tożsamości trygonometrycznych , całkowania przez części oraz pomocniczo z podstawienia
wyprowadzić wzór redukcyjny czy też rekurencyjny
Następnie należałoby rozpatrzeć dwa przypadki początkowe rekurencji
Po rozpoznaniu rekurencji którą otrzymaliśmy będziemy mieli wynik
Korzystając z tożsamości trygonometrycznych , całkowania przez części oraz pomocniczo z podstawienia
wyprowadzić wzór redukcyjny czy też rekurencyjny
Następnie należałoby rozpatrzeć dwa przypadki początkowe rekurencji
Po rozpoznaniu rekurencji którą otrzymaliśmy będziemy mieli wynik
Ukryta treść:
Przypomnijmy wzór na sinusa sumy
\(\displaystyle{ \sin{\left( \alpha + \beta \right) }=\sin{\left( \alpha \right) }\cos{\left( \beta \right) }+\cos{\left( \alpha \right) }\sin{\left( \beta \right) }}\)
\(\displaystyle{ \int_{\theta}^{\pi}{\frac{\sin{\left( \left( n+\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} =\int_{\theta}^{\pi}{\frac{\sin{\left( \left(\left( n-\frac{1}{2}\right) + 1\right)t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}\\
\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n+\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} =\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n-\frac{1}{2}\right) t\right) }\cos{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} +\int_{\theta}^{\pi}{\frac{\cos{\left( \left( n-\frac{1}{2}\right) t\right) }\sin{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} \\
}\)
Obliczmy teraz pomocniczo całkę \(\displaystyle{ \int{\frac{\sin{\left( t\right) }}{\sqrt{2\left(\cos{\left( \theta\right) }-\cos{\left( t\right) }\right)}}\mbox{d}t}}\)
\(\displaystyle{ \int{\frac{\sin{\left( t\right) }}{\sqrt{2\left(\cos{\left( \theta\right) }-\cos{\left( t\right) }\right)}}\mbox{d}t}\\
u^2 = 2\left(\cos{\left( \theta\right) }-\cos{\left( t\right) }\right)\\
2u\mbox{d}u = 2\sin{\left( t\right) }\mbox{d}t\\
u\mbox{d}u = \sin{\left( t\right) }\mbox{d}t\\
\int{\frac{u}{u}\mbox{d}u} = u \\
\int{\frac{\sin{\left( t\right) }}{\sqrt{2\left(\cos{\left( \theta\right) }-\cos{\left( t\right) }\right)}}\mbox{d}t} = \sqrt{2\left(\cos{\left( \theta\right) }-\cos{\left( t\right) }\right)}\\
}\)
\(\displaystyle{
\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n+\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} =\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n-\frac{1}{2}\right) t\right) }\cos{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} +\int_{\theta}^{\pi}{\cos{\left( \left( n-\frac{1}{2}\right) t\right) } \cdot \frac{\sin{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} \\
\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n+\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} =\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n-\frac{1}{2}\right) t\right) }\cos{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} +\cos{\left(\left( n-\frac{1}{2}\right) t\right) }\sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) }\biggl|_{\theta}^{\pi}-\int_{\theta}^{\pi}{\left(n-\frac{1}{2}\right)\left(-\sin{\left( n-\frac{1}{2}\right)t}\right)\sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) }\mbox{d}t}\\
\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n+\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} =\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n-\frac{1}{2}\right) t\right) }\cos{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}+\left(n-\frac{1}{2}\right)\int_{\theta}^{\pi}{\sin{\left( \left( n-\frac{1}{2}\right)t \right) }\sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } \mbox{d}t}\\
\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n+\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} =\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n-\frac{1}{2}\right) t\right) }\cos{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}+\left(n-\frac{1}{2}\right)\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n-\frac{1}{2}\right)t \right) }\left(2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) \right)}{\sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) }} \mbox{d}t}\\
\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n+\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} =\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n-\frac{1}{2}\right) t\right) }\cos{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}+\left( 2n-1\right) \cos{\left( \theta\right) }\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n-\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} -\left(2n-1\right)\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n-\frac{1}{2}\right) t\right) }\cos{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}\\
\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n+\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} =\left(1-\left(2n-1\right)\right)\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n-\frac{1}{2}\right) t\right) }\cos{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}+\left( 2n-1\right) \cos{\left( \theta\right) }\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n-\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}\\
\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n+\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}=\left( 2n-1\right) \cos{\left( \theta\right) }\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n-\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}-\left( n-1\right)\int_{\theta}^{\pi}{\frac{2\sin{\left( \left( n-\frac{1}{2}\right) t\right) }\cos{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} \\
}\)
\(\displaystyle{ \sin{\left( \alpha + \beta \right) }=\sin{\left( \alpha \right) }\cos{\left( \beta \right) }+\cos{\left( \alpha \right) }\sin{\left( \beta \right) }\\
\sin{\left( \alpha - \beta \right) }=\sin{\left( \alpha \right) }\cos{\left( \beta \right) }-\cos{\left( \alpha \right) }\sin{\left( \beta \right) }\\
\sin{\left( \alpha + \beta \right) }+\sin{\left( \alpha - \beta \right) } = 2\sin{\left( \alpha \right) }\cos{\left( \beta \right) }\\
}\)
\(\displaystyle{
\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n+\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}=\left( 2n-1\right) \cos{\left( \theta\right) }\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n-\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}-\left( n-1\right)\int_{\theta}^{\pi}{\frac{2\sin{\left( \left( n-\frac{1}{2}\right) t\right) }\cos{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} \\
\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n+\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}=\left( 2n-1\right) \cos{\left( \theta\right) }\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n-1+\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}-\left( n-1\right)\int_{\theta}^{\pi}{\frac{\left(\sin{\left( \left( n+\frac{1}{2}\right) t\right) } + \sin{\left( \left( n-2+\frac{1}{2}\right) t\right) }\right)}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} \\
\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n+\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}=\left( 2n-1\right) \cos{\left( \theta\right) }\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n-1+\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}-\left( n-1\right)\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n+\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}-\left( n-1\right)\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n-2+\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}\\
n\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n+\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}=\left( 2n-1\right) \cos{\left( \theta\right) }\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n-1+\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}-\left( n-1\right)\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n - 2 +\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}\\
}\)
Mamy równanie rekurencyjne obliczny teraz warunki początkowe dla ten rekurencji
Rozważmy przypadek gdy \(\displaystyle{ n = 0}\)
\(\displaystyle{
\int_{\theta}^{\pi}{\frac{\sin{\left(\frac{1}{2} t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}=\int_{\theta}^{\pi}{\frac{\sin{\left( \frac{t}{2}\right) }}{\sqrt{2\cos{\left(\theta\right)}-2\left(\cos^2{\left(\frac{1}{2} t\right) }-\sin^2{\left(\frac{1}{2} t\right) }\right)}}\mbox{d}t}\\
\int_{\theta}^{\pi}{\frac{\sin{\left(\frac{1}{2} t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}=\int_{\theta}^{\pi}{\frac{\sin{\left( \frac{t}{2}\right) }}{\sqrt{2\cos{\left(\theta\right)}-2\left(\cos^2{\left(\frac{1}{2} t\right) }-\left(1-\cos^2{\left(\frac{1}{2} t\right) }\right)\right)}}\mbox{d}t}\\
\int_{\theta}^{\pi}{\frac{\sin{\left(\frac{1}{2} t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}=\int_{\theta}^{\pi}{\frac{\sin{\left( \frac{t}{2}\right) }}{\sqrt{2\cos{\left(\theta\right)} + 2 - 4\cos^2{\left(\frac{1}{2} t\right) }}}\mbox{d}t}\\
\sqrt{2\cos{\left( \theta\right) + 2}} \cdot u = 2\cos{\left( \frac{t}{2} \right) }\\
\sqrt{2\cos{\left( \theta\right) + 2}}\mbox{d}u = -2 \cdot \frac{1}{2}\sin{\left( \frac{t}{2} \right) }\mbox{d}t\\
\sqrt{2\cos{\left( \theta\right) + 2}}\mbox{d}u = -\sin{\left( \frac{t}{2} \right) }\mbox{d}t\\
-\sqrt{2\cos{\left( \theta\right) + 2}}\mbox{d}u = \sin{\left( \frac{t}{2} \right) }\mbox{d}t\\
t = \theta\\
\sqrt{2\cos{\left( \theta\right) }+2} \cdot u = 2\cos{\left( \frac{\theta}{2} \right) }\\
2 \cdot \sqrt{ \frac{1+\cos{\left( \theta\right) }}{2} } \cdot u = 2\cos{\left( \frac{\theta}{2} \right) }\\
\left|\cos{\left( \frac{\theta}{2} \right) } \right| \cdot u = \cos{\left( \frac{\theta}{2} \right) }\\
u = \frac{\cos{\left( \frac{\theta}{2} \right) }}{\left|\cos{\left( \frac{\theta}{2} \right) } \right|}\\
t = \pi \\
\sqrt{2\cos{\left( \theta\right) + 2}} \cdot u = 2\cos{\left( \frac{\pi}{2} \right) }\\
\sqrt{2\cos{\left( \theta\right) + 2}} \cdot u =0\\
u = 0\\
\int_{\theta}^{\pi}{\frac{\sin{\left(\frac{1}{2} t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}=-\int_{\frac{\cos{\left( \frac{\theta}{2} \right) }}{\left|\cos{\left( \frac{\theta}{2} \right) } \right|}}^{0}{\sqrt{2\cos{\left( \theta\right) + 2}} \cdot \frac{1}{\sqrt{\left(2\cos{\left( \theta\right) }+2\right) -\left(2\cos{\left( \theta\right) }+2\right)u^2 }}\mbox{d}u}\\
\int_{\theta}^{\pi}{\frac{\sin{\left(\frac{1}{2} t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}=\int_{0}^{\frac{\cos{\left( \frac{\theta}{2} \right) }}{\left|\cos{\left( \frac{\theta}{2} \right) } \right|}}{\frac{1}{\sqrt{1-u^2}}\mbox{d}u}\\
\int_{\theta}^{\pi}{\frac{\sin{\left(\frac{1}{2} t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}=\arcsin{\left( \frac{\cos{\left( \frac{\theta}{2} \right) }}{\left| \cos{\left( \frac{\theta}{2} \right) }\right| } \right) }\\
}\)
Jeżeli przyjmiemy że \(\displaystyle{ \cos{\left(\frac{\theta}{2}\right)} > 0}\) to otrzymamy
\(\displaystyle{
\int_{\theta}^{\pi}{\frac{\sin{\left(\frac{1}{2} t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}=\frac{\pi}{2}
}\)
Rozważmy przypadek gdy \(\displaystyle{ n = 1}\)
\(\displaystyle{
\int_{\theta}^{\pi}{\frac{\sin{\left(\frac{3}{2} t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} = \int_{\theta}^{\pi}{\frac{\sin{\left(\frac{1}{2} t\right) }\cos{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}+\int_{\theta}^{\pi}{\frac{\cos{\left(\frac{1}{2} t\right) }\sin{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}\\
\int_{\theta}^{\pi}{\frac{\sin{\left(\frac{3}{2} t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} = \int_{\theta}^{\pi}{\frac{\sin{\left(\frac{1}{2} t\right) }\cos{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}+\cos{\left( \frac{t}{2} \right) } \cdot \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } \biggl|_{\theta}^{\pi} - \int_{\theta}^{\pi}{\left(-\frac{1}{2}\sin{\left( \frac{t}{2} \right) }\right)\cdot \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) }\mbox{d}t}\\
\int_{\theta}^{\pi}{\frac{\sin{\left(\frac{3}{2} t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} = \int_{\theta}^{\pi}{\frac{\sin{\left(\frac{1}{2} t\right) }\cos{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}+\frac{1}{2}\int_{\theta}^{\pi}{\sin{\left( \frac{t}{2} \right) } \cdot \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } \mbox{d}t}\\
\int_{\theta}^{\pi}{\frac{\sin{\left(\frac{3}{2} t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} = \int_{\theta}^{\pi}{\frac{\sin{\left(\frac{1}{2} t\right) }\cos{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}+\frac{1}{2}\int_{\theta}^{\pi}{\sin{\left( \frac{t}{2} \right) } \cdot \frac{2\cos{\left( \theta\right) } - 2\cos{\left( t\right) }}{\sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) }} \mbox{d}t}\\
\int_{\theta}^{\pi}{\frac{\sin{\left(\frac{3}{2} t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} = \int_{\theta}^{\pi}{\frac{\sin{\left(\frac{1}{2} t\right) }\cos{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}+\cos{\left( \theta\right) }\int_{\theta}^{\pi}{\frac{\sin{\left(\frac{3}{2} t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}-\int_{\theta}^{\pi}{\frac{\sin{\left(\frac{1}{2} t\right) }\cos{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}\\
\int_{\theta}^{\pi}{\frac{\sin{\left(\frac{3}{2} t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} = \cos{\left( \theta\right) }\int_{\theta}^{\pi}{\frac{\sin{\left(\frac{1}{2} t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}\\
\int_{\theta}^{\pi}{\frac{\sin{\left(\frac{3}{2} t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} = \cos{\left( \theta\right) }\cdot \arcsin{\left( \frac{\cos{\left( \frac{\theta}{2} \right) }}{\left| \cos{\left( \frac{\theta}{2} \right) }\right| } \right) }\\
}\)
Jeżeli przyjmiemy że \(\displaystyle{ \cos{\left(\frac{\theta}{2}\right)} > 0}\) to otrzymamy
\(\displaystyle{
\int_{\theta}^{\pi}{\frac{\sin{\left(\frac{3}{2} t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}=\frac{\pi}{2}\cdot\cos{\left( \theta\right) }\\
}\)
Dla \(\displaystyle{ \cos{\left(\frac{\theta}{2}\right)} > 0 }\) otrzymamy następującą rekurencję
\(\displaystyle{
\Large
{
I_{n} = \begin{cases}\frac{\pi}{2}, \qquad n=0\\\frac{\pi}{2} \cdot \cos{\left( \theta\right) }, \qquad n=1\\\frac{2n-1}{n}\cdot \cos{\left( \theta\right) } \cdot I_{n-1} - \frac{n-1}{n}I_{n-2}, \qquad n \ge 2 \end{cases}\\
}
}\)
\(\displaystyle{ \sin{\left( \alpha + \beta \right) }=\sin{\left( \alpha \right) }\cos{\left( \beta \right) }+\cos{\left( \alpha \right) }\sin{\left( \beta \right) }}\)
\(\displaystyle{ \int_{\theta}^{\pi}{\frac{\sin{\left( \left( n+\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} =\int_{\theta}^{\pi}{\frac{\sin{\left( \left(\left( n-\frac{1}{2}\right) + 1\right)t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}\\
\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n+\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} =\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n-\frac{1}{2}\right) t\right) }\cos{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} +\int_{\theta}^{\pi}{\frac{\cos{\left( \left( n-\frac{1}{2}\right) t\right) }\sin{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} \\
}\)
Obliczmy teraz pomocniczo całkę \(\displaystyle{ \int{\frac{\sin{\left( t\right) }}{\sqrt{2\left(\cos{\left( \theta\right) }-\cos{\left( t\right) }\right)}}\mbox{d}t}}\)
\(\displaystyle{ \int{\frac{\sin{\left( t\right) }}{\sqrt{2\left(\cos{\left( \theta\right) }-\cos{\left( t\right) }\right)}}\mbox{d}t}\\
u^2 = 2\left(\cos{\left( \theta\right) }-\cos{\left( t\right) }\right)\\
2u\mbox{d}u = 2\sin{\left( t\right) }\mbox{d}t\\
u\mbox{d}u = \sin{\left( t\right) }\mbox{d}t\\
\int{\frac{u}{u}\mbox{d}u} = u \\
\int{\frac{\sin{\left( t\right) }}{\sqrt{2\left(\cos{\left( \theta\right) }-\cos{\left( t\right) }\right)}}\mbox{d}t} = \sqrt{2\left(\cos{\left( \theta\right) }-\cos{\left( t\right) }\right)}\\
}\)
\(\displaystyle{
\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n+\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} =\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n-\frac{1}{2}\right) t\right) }\cos{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} +\int_{\theta}^{\pi}{\cos{\left( \left( n-\frac{1}{2}\right) t\right) } \cdot \frac{\sin{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} \\
\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n+\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} =\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n-\frac{1}{2}\right) t\right) }\cos{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} +\cos{\left(\left( n-\frac{1}{2}\right) t\right) }\sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) }\biggl|_{\theta}^{\pi}-\int_{\theta}^{\pi}{\left(n-\frac{1}{2}\right)\left(-\sin{\left( n-\frac{1}{2}\right)t}\right)\sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) }\mbox{d}t}\\
\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n+\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} =\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n-\frac{1}{2}\right) t\right) }\cos{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}+\left(n-\frac{1}{2}\right)\int_{\theta}^{\pi}{\sin{\left( \left( n-\frac{1}{2}\right)t \right) }\sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } \mbox{d}t}\\
\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n+\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} =\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n-\frac{1}{2}\right) t\right) }\cos{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}+\left(n-\frac{1}{2}\right)\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n-\frac{1}{2}\right)t \right) }\left(2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) \right)}{\sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) }} \mbox{d}t}\\
\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n+\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} =\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n-\frac{1}{2}\right) t\right) }\cos{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}+\left( 2n-1\right) \cos{\left( \theta\right) }\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n-\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} -\left(2n-1\right)\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n-\frac{1}{2}\right) t\right) }\cos{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}\\
\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n+\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} =\left(1-\left(2n-1\right)\right)\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n-\frac{1}{2}\right) t\right) }\cos{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}+\left( 2n-1\right) \cos{\left( \theta\right) }\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n-\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}\\
\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n+\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}=\left( 2n-1\right) \cos{\left( \theta\right) }\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n-\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}-\left( n-1\right)\int_{\theta}^{\pi}{\frac{2\sin{\left( \left( n-\frac{1}{2}\right) t\right) }\cos{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} \\
}\)
\(\displaystyle{ \sin{\left( \alpha + \beta \right) }=\sin{\left( \alpha \right) }\cos{\left( \beta \right) }+\cos{\left( \alpha \right) }\sin{\left( \beta \right) }\\
\sin{\left( \alpha - \beta \right) }=\sin{\left( \alpha \right) }\cos{\left( \beta \right) }-\cos{\left( \alpha \right) }\sin{\left( \beta \right) }\\
\sin{\left( \alpha + \beta \right) }+\sin{\left( \alpha - \beta \right) } = 2\sin{\left( \alpha \right) }\cos{\left( \beta \right) }\\
}\)
\(\displaystyle{
\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n+\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}=\left( 2n-1\right) \cos{\left( \theta\right) }\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n-\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}-\left( n-1\right)\int_{\theta}^{\pi}{\frac{2\sin{\left( \left( n-\frac{1}{2}\right) t\right) }\cos{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} \\
\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n+\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}=\left( 2n-1\right) \cos{\left( \theta\right) }\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n-1+\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}-\left( n-1\right)\int_{\theta}^{\pi}{\frac{\left(\sin{\left( \left( n+\frac{1}{2}\right) t\right) } + \sin{\left( \left( n-2+\frac{1}{2}\right) t\right) }\right)}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} \\
\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n+\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}=\left( 2n-1\right) \cos{\left( \theta\right) }\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n-1+\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}-\left( n-1\right)\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n+\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}-\left( n-1\right)\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n-2+\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}\\
n\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n+\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}=\left( 2n-1\right) \cos{\left( \theta\right) }\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n-1+\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}-\left( n-1\right)\int_{\theta}^{\pi}{\frac{\sin{\left( \left( n - 2 +\frac{1}{2}\right) t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}\\
}\)
Mamy równanie rekurencyjne obliczny teraz warunki początkowe dla ten rekurencji
Rozważmy przypadek gdy \(\displaystyle{ n = 0}\)
\(\displaystyle{
\int_{\theta}^{\pi}{\frac{\sin{\left(\frac{1}{2} t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}=\int_{\theta}^{\pi}{\frac{\sin{\left( \frac{t}{2}\right) }}{\sqrt{2\cos{\left(\theta\right)}-2\left(\cos^2{\left(\frac{1}{2} t\right) }-\sin^2{\left(\frac{1}{2} t\right) }\right)}}\mbox{d}t}\\
\int_{\theta}^{\pi}{\frac{\sin{\left(\frac{1}{2} t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}=\int_{\theta}^{\pi}{\frac{\sin{\left( \frac{t}{2}\right) }}{\sqrt{2\cos{\left(\theta\right)}-2\left(\cos^2{\left(\frac{1}{2} t\right) }-\left(1-\cos^2{\left(\frac{1}{2} t\right) }\right)\right)}}\mbox{d}t}\\
\int_{\theta}^{\pi}{\frac{\sin{\left(\frac{1}{2} t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}=\int_{\theta}^{\pi}{\frac{\sin{\left( \frac{t}{2}\right) }}{\sqrt{2\cos{\left(\theta\right)} + 2 - 4\cos^2{\left(\frac{1}{2} t\right) }}}\mbox{d}t}\\
\sqrt{2\cos{\left( \theta\right) + 2}} \cdot u = 2\cos{\left( \frac{t}{2} \right) }\\
\sqrt{2\cos{\left( \theta\right) + 2}}\mbox{d}u = -2 \cdot \frac{1}{2}\sin{\left( \frac{t}{2} \right) }\mbox{d}t\\
\sqrt{2\cos{\left( \theta\right) + 2}}\mbox{d}u = -\sin{\left( \frac{t}{2} \right) }\mbox{d}t\\
-\sqrt{2\cos{\left( \theta\right) + 2}}\mbox{d}u = \sin{\left( \frac{t}{2} \right) }\mbox{d}t\\
t = \theta\\
\sqrt{2\cos{\left( \theta\right) }+2} \cdot u = 2\cos{\left( \frac{\theta}{2} \right) }\\
2 \cdot \sqrt{ \frac{1+\cos{\left( \theta\right) }}{2} } \cdot u = 2\cos{\left( \frac{\theta}{2} \right) }\\
\left|\cos{\left( \frac{\theta}{2} \right) } \right| \cdot u = \cos{\left( \frac{\theta}{2} \right) }\\
u = \frac{\cos{\left( \frac{\theta}{2} \right) }}{\left|\cos{\left( \frac{\theta}{2} \right) } \right|}\\
t = \pi \\
\sqrt{2\cos{\left( \theta\right) + 2}} \cdot u = 2\cos{\left( \frac{\pi}{2} \right) }\\
\sqrt{2\cos{\left( \theta\right) + 2}} \cdot u =0\\
u = 0\\
\int_{\theta}^{\pi}{\frac{\sin{\left(\frac{1}{2} t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}=-\int_{\frac{\cos{\left( \frac{\theta}{2} \right) }}{\left|\cos{\left( \frac{\theta}{2} \right) } \right|}}^{0}{\sqrt{2\cos{\left( \theta\right) + 2}} \cdot \frac{1}{\sqrt{\left(2\cos{\left( \theta\right) }+2\right) -\left(2\cos{\left( \theta\right) }+2\right)u^2 }}\mbox{d}u}\\
\int_{\theta}^{\pi}{\frac{\sin{\left(\frac{1}{2} t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}=\int_{0}^{\frac{\cos{\left( \frac{\theta}{2} \right) }}{\left|\cos{\left( \frac{\theta}{2} \right) } \right|}}{\frac{1}{\sqrt{1-u^2}}\mbox{d}u}\\
\int_{\theta}^{\pi}{\frac{\sin{\left(\frac{1}{2} t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}=\arcsin{\left( \frac{\cos{\left( \frac{\theta}{2} \right) }}{\left| \cos{\left( \frac{\theta}{2} \right) }\right| } \right) }\\
}\)
Jeżeli przyjmiemy że \(\displaystyle{ \cos{\left(\frac{\theta}{2}\right)} > 0}\) to otrzymamy
\(\displaystyle{
\int_{\theta}^{\pi}{\frac{\sin{\left(\frac{1}{2} t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}=\frac{\pi}{2}
}\)
Rozważmy przypadek gdy \(\displaystyle{ n = 1}\)
\(\displaystyle{
\int_{\theta}^{\pi}{\frac{\sin{\left(\frac{3}{2} t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} = \int_{\theta}^{\pi}{\frac{\sin{\left(\frac{1}{2} t\right) }\cos{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}+\int_{\theta}^{\pi}{\frac{\cos{\left(\frac{1}{2} t\right) }\sin{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}\\
\int_{\theta}^{\pi}{\frac{\sin{\left(\frac{3}{2} t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} = \int_{\theta}^{\pi}{\frac{\sin{\left(\frac{1}{2} t\right) }\cos{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}+\cos{\left( \frac{t}{2} \right) } \cdot \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } \biggl|_{\theta}^{\pi} - \int_{\theta}^{\pi}{\left(-\frac{1}{2}\sin{\left( \frac{t}{2} \right) }\right)\cdot \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) }\mbox{d}t}\\
\int_{\theta}^{\pi}{\frac{\sin{\left(\frac{3}{2} t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} = \int_{\theta}^{\pi}{\frac{\sin{\left(\frac{1}{2} t\right) }\cos{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}+\frac{1}{2}\int_{\theta}^{\pi}{\sin{\left( \frac{t}{2} \right) } \cdot \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } \mbox{d}t}\\
\int_{\theta}^{\pi}{\frac{\sin{\left(\frac{3}{2} t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} = \int_{\theta}^{\pi}{\frac{\sin{\left(\frac{1}{2} t\right) }\cos{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}+\frac{1}{2}\int_{\theta}^{\pi}{\sin{\left( \frac{t}{2} \right) } \cdot \frac{2\cos{\left( \theta\right) } - 2\cos{\left( t\right) }}{\sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) }} \mbox{d}t}\\
\int_{\theta}^{\pi}{\frac{\sin{\left(\frac{3}{2} t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} = \int_{\theta}^{\pi}{\frac{\sin{\left(\frac{1}{2} t\right) }\cos{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}+\cos{\left( \theta\right) }\int_{\theta}^{\pi}{\frac{\sin{\left(\frac{3}{2} t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}-\int_{\theta}^{\pi}{\frac{\sin{\left(\frac{1}{2} t\right) }\cos{\left( t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}\\
\int_{\theta}^{\pi}{\frac{\sin{\left(\frac{3}{2} t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} = \cos{\left( \theta\right) }\int_{\theta}^{\pi}{\frac{\sin{\left(\frac{1}{2} t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}\\
\int_{\theta}^{\pi}{\frac{\sin{\left(\frac{3}{2} t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t} = \cos{\left( \theta\right) }\cdot \arcsin{\left( \frac{\cos{\left( \frac{\theta}{2} \right) }}{\left| \cos{\left( \frac{\theta}{2} \right) }\right| } \right) }\\
}\)
Jeżeli przyjmiemy że \(\displaystyle{ \cos{\left(\frac{\theta}{2}\right)} > 0}\) to otrzymamy
\(\displaystyle{
\int_{\theta}^{\pi}{\frac{\sin{\left(\frac{3}{2} t\right) }}{ \sqrt{2\left( \cos{\left( \theta\right) } - \cos{\left( t\right) }\right) } }\mbox{d}t}=\frac{\pi}{2}\cdot\cos{\left( \theta\right) }\\
}\)
Dla \(\displaystyle{ \cos{\left(\frac{\theta}{2}\right)} > 0 }\) otrzymamy następującą rekurencję
\(\displaystyle{
\Large
{
I_{n} = \begin{cases}\frac{\pi}{2}, \qquad n=0\\\frac{\pi}{2} \cdot \cos{\left( \theta\right) }, \qquad n=1\\\frac{2n-1}{n}\cdot \cos{\left( \theta\right) } \cdot I_{n-1} - \frac{n-1}{n}I_{n-2}, \qquad n \ge 2 \end{cases}\\
}
}\)