Integrals

Real Analysis
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Tolaso J Kos
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Integrals

#1

Post by Tolaso J Kos »

(a) Prove that: \( \displaystyle \int_{0}^{\infty}\sin^2 \left ( \pi\left ( x+\frac{1}{x} \right ) \right )\, dx \) does not exist.

(b) Prove that: \( \displaystyle \lim_{k\rightarrow +\infty}\int_{0}^{\infty}\frac{dx}{1+kx^{10}}=0 \).
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Re: Integrals

#2

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Replied by ex-member aziiri:


(a) We prove that the integral over \((0,+\infty)\) does not exist : \[\int_1^{\infty} \sin^2 \left ( \pi\left ( x+\frac{1}{x} \right ) \right ) \ \mathrm{d}x \overset{x+x^{-1}=t}{=} =\frac{1}{2} \int_2^{\infty} \sin^2 (\pi t) \left(1+\frac{t}{\sqrt{t^2-4}} \right) \ \mathrm{d}t\geq \frac{1}{2} \int_2^{\infty} \sin^2 (\pi t)\ \mathrm{d}t\] The latter is clearly divergent to \(+\infty\).

(b) The integrand is less than \(\frac{1}{1+x^{10}}\) (which is integrable) by absolute value, then we can interchange the limit-integral order to get \(0\) as the result.
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