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Math Help - nice integral

  1. #1
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    nice integral

    evaluate \int_0^\pi\ln(1+\alpha\cos x)\,dx
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  2. #2
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    \int_{0}^{\pi }{\ln (1+\alpha \cos x)\,dx}=\int_{0}^{\pi }{\int_{0}^{\alpha }{\frac{\cos x}{1+y\cos x}\,dy}\,dx}, that leads to \pi \int_{0}^{\alpha }{\frac{\sqrt{1-y^{2}}-1}{y\sqrt{1-y^{2}}}\,dy}, which is easily killed by makin' the substitution u=\ln \left( 1+\sqrt{1-y^{2}} \right). Finally, your "nice" integral equals \pi \cdot \ln \frac{1+\sqrt{1-\alpha ^{2}}}{2}, which holds for |\alpha|\le1 and we're done.
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  3. #3
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Ajreb View Post
    evaluate \int_0^\pi\ln(1+\alpha\cos x)\,dx
    Exactly the same method as Krizalid's just a slightly different start...it is how I solved this integral the first time I saw it... sometimes if its not apparent to you just remember that if you ever see an integral with a second variable expression in it try letting it equal to its own function.

    J\left(\alpha\right)=\int_0^{\pi}\ln\left(1+\alpha  \cos(x)\right)dx

    So

    \begin{aligned}J'\left(\alpha\right)&=\frac{d}{d\a  lpha}\int_0^{\pi}\ln\left(1+\alpha\cos(x)\right)dx  \\<br />
&=\int_0^{\pi}\frac{\partial}{\partial\alpha}\ln\l  eft(1+\alpha\cos(x)\right)dx\\<br />
&=\int_0^{\pi}\frac{\cos(x)}{\alpha\cos(x)+1}dx<br />
\end{aligned}

    So the same method as Krizalid's just a different way of seeing it.
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