MHB Evaluating Integrals Involving Trig Functions

AI Thread Summary
The discussion focuses on evaluating two integrals involving trigonometric functions. For the first integral, it is shown that by using properties of definite integrals, the result simplifies to I = π². The second integral is approached similarly, leading to the conclusion that I = π²/16 after appropriate substitutions and transformations. Both integrals demonstrate the utility of symmetry and substitution in simplifying complex expressions. The thread emphasizes the importance of understanding integral properties in solving trigonometric integrals effectively.
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Evaluate:

1. $\displaystyle \int_0^{\displaystyle 2\pi} \frac{x \sin^{2n}(x)}{\sin^{2n}(x)+\cos^{2n}(x)}dx$, $n>0$

2. $\displaystyle \int_0^{ \displaystyle \pi \over \displaystyle 2} \frac{x \sin x \cos x}{\sin^{4}(x)+\cos^{4}(x)}dx$
 
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In 1) - is $n$ a positive integer?
 
Jester said:
In 1) - is $n$ a positive integer?

Yes! Sorry, I should I have mentioned at the beginning.
 
Here is a hint: Try using properties of definite integrals.
 
Putting $x \mapsto 2\pi-x$ we have:$\displaystyle I = \int_{0}^{2\pi}\frac{x\sin^{2n}{x}}{\sin^{2n}{x}+ \cos^{2n}{x}}\;{dx} = \int_{0}^{2\pi}\frac{(2\pi-x)\sin^{2n}{x}}{\sin^{2n}{x}+\cos^{2n}{x}}\;{dx} = 2\pi\int_{0}^{2\pi}\frac{\sin^{2n}{x}}{\sin^{2n}{x}+\cos^{2n}{x}}\;{dx}-I$.Thus $ \displaystyle I = \pi\int_{0}^{2\pi}\frac{\sin^{2n}{x}}{\sin^{2n}{x}+\cos^{2n}{x}}\;{dx} $. Now, write this from $[0, ~ \pi/2]$ and $[\pi/2, ~ 2\pi]$, then:$\displaystyle I = \pi\int_{0}^{\pi/2}\frac{\sin^{2n}{x}}{\sin^{2n}{x}+\cos^{2n}{x}}\;{dx} +\pi\int_{\pi/2}^{2\pi}\frac{\sin^{2n}{x}}{\sin^{2n}{x}+\cos^{2n}{x}}\;{dx}$. Put $x\mapsto \frac{\pi}{2}-x$ and $x\mapsto \frac{5\pi}{2}-x$, then:$\displaystyle I = \pi\int_{0}^{\pi/2}\frac{\cos^{2n}{x}}{\sin^{2n}{x}+\cos^{2n}{x}}\;{dx} +\pi\int_{\pi/2}^{2\pi}\frac{\cos^{2n}{x}}{\sin^{2n}{x}+\cos^{2n}{x}}\;{dx}$. Add this and the above, then we've:

$\displaystyle 2I = 2\pi^2 \Rightarrow \boxed{I = \pi^2}$. I'll leave the second one for others to try.
 
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Sherlock said:
Putting $x \mapsto 2\pi-x$ we have:$\displaystyle I = \int_{0}^{2\pi}\frac{x\sin^{2n}{x}}{\sin^{2n}{x}+ \cos^{2n}{x}}\;{dx} = \int_{0}^{2\pi}\frac{(2\pi-x)\sin^{2n}{x}}{\sin^{2n}{x}+\cos^{2n}{x}}\;{dx} = 2\pi\int_{0}^{2\pi}\frac{\sin^{2n}{x}}{\sin^{2n}{x}+\cos^{2n}{x}}\;{dx}-I$.Thus $ \displaystyle I = \pi\int_{0}^{2\pi}\frac{\sin^{2n}{x}}{\sin^{2n}{x}+\cos^{2n}{x}}\;{dx} $. Now, write this from $[0, ~ \pi/2]$ and $[\pi/2, ~ 2\pi]$, then:$\displaystyle I = \pi\int_{0}^{\pi/2}\frac{\sin^{2n}{x}}{\sin^{2n}{x}+\cos^{2n}{x}}\;{dx} +\pi\int_{\pi/2}^{2\pi}\frac{\sin^{2n}{x}}{\sin^{2n}{x}+\cos^{2n}{x}}\;{dx}$. Put $x\mapsto \frac{\pi}{2}-x$ and $x\mapsto \frac{5\pi}{2}-x$, then:$\displaystyle I = \pi\int_{0}^{\pi/2}\frac{\cos^{2n}{x}}{\sin^{2n}{x}+\cos^{2n}{x}}\;{dx} +\pi\int_{\pi/2}^{2\pi}\frac{\cos^{2n}{x}}{\sin^{2n}{x}+\cos^{2n}{x}}\;{dx}$. Add this and the above, then we've:

$\displaystyle 2I = 2\pi^2 \Rightarrow \boxed{I = \pi^2}$. I'll leave the second one for others to try.

Yeah, that's right.
 
Solution to 2.

Let $\displaystyle I= \int_0^{\pi / 2}\frac{x \sin x \cos x}{\sin^4 x + \cos^4 x}dx$

By a property of definite integrals, we get

$\displaystyle I= \int_0^{\pi /2}\dfrac{\displaystyle \left( \dfrac{\pi}{2} -x\right) \sin \left( \dfrac{\pi}{2} -x\right) \cos \left( \dfrac{\pi}{2} -x\right)}{ \sin^{4}\left( \dfrac{\pi}{2} -x\right) + \cos^4 \left( \dfrac{\pi}{2} -x\right)}dx = \int_0^{\pi /2}\dfrac{\displaystyle \left( \dfrac{\pi}{2} -x\right) \sin x \cos x}{ \sin^{4}x+ \cos^4 x }dx ={\pi \over 2}\int_0^{\pi / 2}\frac{ \sin x \cos x}{\sin^4 x + \cos^4 x}dx -I$

$\displaystyle \therefore \ \ 2I= \frac{\pi}{2} \int_0^{\pi/2}\frac{\sin x \cos x}{\sin^4 x + \cos ^4 x}dx =\frac{\pi}{2} \int_0^{\pi /2}\frac{\tan x \sec^2 x}{1+\tan^4 x }dx$

Substitute $t= \tan^2 x$ and $dt = 2 \tan x \sec^2 x \ dx$.

$\displaystyle 2I= \frac{\pi}{4}\int_{0}^{\infty}\frac{1}{1+t^2}dt = \frac{\pi^2}{8}$

$\displaystyle \therefore \ \ I= \frac{\pi^2}{16}$
 
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