Win_odd Dhamnekar
Junior Member
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- Aug 14, 2018
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Hello,
How to find [MATH]\displaystyle\int_0^{\pi} \frac{dx}{(5+3*cos(x))^3}[/MATH] and[MATH]\displaystyle\int_0^{\pi}\frac{sin^2(x)}{(5+3*cos(x))^3}[/MATH] by differentiating under the integral sign via Feynman's trick? I am given the general result that [MATH]\displaystyle\int_0^{\pi}\frac{dx}{(a+b*cos(x))}=\frac{\pi}{\sqrt{(a^2-b^2)}}[/MATH]
Solution:-
If 0<a<b we have,
[MATH]\frac12\displaystyle\int_0^{2\pi}\frac{d\theta}{(b+a*cos(\theta))^2}d\theta=\frac{\pi b}{(b^2-a^2)^{\frac32})} …(1)[/MATH]
By differentiating both sides of (1) with respect to b,
[MATH]\displaystyle\int_0^{2\pi}\frac{d\theta}{(b+a*cos(\theta))^3}=\frac{\pi*(a^2+2*b^2)}{(b^2-a^2)^{\frac52}}…(2)[/MATH]
and by differentiating both sides of (1) with respect to a,
[MATH]\displaystyle\int_0^{2\pi}\frac{cos(\theta)d\theta}{(b+a cos(\theta))^3}=\frac{3\pi a *b}{(b^2-a^2)^{\frac52}}…(3)[/MATH]
Now, how to proceed further, by using (2) and (3) above , to compute the required integrals?
If any member knows the correct answer, may reply with correct answer. I know the correct answer is [MATH]\frac{59*\pi}{2048} and\frac{\pi}{128}[/MATH] respectively................... edited
How to find [MATH]\displaystyle\int_0^{\pi} \frac{dx}{(5+3*cos(x))^3}[/MATH] and[MATH]\displaystyle\int_0^{\pi}\frac{sin^2(x)}{(5+3*cos(x))^3}[/MATH] by differentiating under the integral sign via Feynman's trick? I am given the general result that [MATH]\displaystyle\int_0^{\pi}\frac{dx}{(a+b*cos(x))}=\frac{\pi}{\sqrt{(a^2-b^2)}}[/MATH]
Solution:-
If 0<a<b we have,
[MATH]\frac12\displaystyle\int_0^{2\pi}\frac{d\theta}{(b+a*cos(\theta))^2}d\theta=\frac{\pi b}{(b^2-a^2)^{\frac32})} …(1)[/MATH]
By differentiating both sides of (1) with respect to b,
[MATH]\displaystyle\int_0^{2\pi}\frac{d\theta}{(b+a*cos(\theta))^3}=\frac{\pi*(a^2+2*b^2)}{(b^2-a^2)^{\frac52}}…(2)[/MATH]
and by differentiating both sides of (1) with respect to a,
[MATH]\displaystyle\int_0^{2\pi}\frac{cos(\theta)d\theta}{(b+a cos(\theta))^3}=\frac{3\pi a *b}{(b^2-a^2)^{\frac52}}…(3)[/MATH]
Now, how to proceed further, by using (2) and (3) above , to compute the required integrals?
If any member knows the correct answer, may reply with correct answer. I know the correct answer is [MATH]\frac{59*\pi}{2048} and\frac{\pi}{128}[/MATH] respectively................... edited
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