I don't understand why I solve the integration in two different ways and get two different answers!!(adsbygoogle = window.adsbygoogle || []).push({});

To find:

[tex]\int_0^{\pi} P_1^1(cos \theta) sin \theta d \theta [/tex]

1) Solve in [itex] \theta[/itex]

[tex] P_1(cos \theta) = cos \theta \;\Rightarrow \; P_1^1(cos \theta)= -sin \theta [/tex]

[tex]\int_0^{\pi} P_1^1(cos \theta) sin \theta d \theta = -\int_0^{\pi}sin^2 \theta d \theta = -\frac{\pi}{2}[/tex]

2) Let [tex]s=cos \theta \;\Rightarrow \; d\theta = \frac{ds}{-sin \theta} [/tex]

[tex]\int_0^{\pi} P_1^1(cos \theta) sin \theta d \theta = -\int _1^{-1} P_1^1(s)ds[/tex]

[tex]P_1(s)=s \;\Rightarrow P_1^1(s)=1[/tex]

[tex]\int_0^{\pi} P_1^1(cos \theta) sin \theta d \theta = -\int _1^{-1} P_1^1(s)ds = -s|_1^{-1} = 2[/tex]

You see the two method yield two answers!!! I know it should yield the same answer, the book show how to solve the problems in 2) form. Please tell me what did I do wrong.

thanks

Alan

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# Please help in integration of Associate Legendre function

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