Integral equality
https://imgur.com/u0rL1EV
Is it true that
\[\int_{-\frac{\pi}{2}}^{\pi}\frac{\sin ^3 (\pi x)}{1+x^{10}}dx=\int_{\frac{\pi}{2}}^{\pi}\frac{\sin ^3 (\pi x)}{1+x^{10}}dx?\]
Is it true that
\[\int_{-\frac{\pi}{2}}^{\pi}\frac{\sin ^3 (\pi x)}{1+x^{10}}dx=\int_{\frac{\pi}{2}}^{\pi}\frac{\sin ^3 (\pi x)}{1+x^{10}}dx?\]
![Ii Skils](https://matchmaticians.com/storage/user/104826/thumb/AGNmyxZ4-zRsgEDK67WkaKp7m8fnCT_ioNK-011Y7wT97Q-avatar-512.jpg)
1
1 Answer
The answer is true. We have
\[\int_{-\frac{\pi}{2}}^{\pi}\frac{\sin ^3 (\pi x)}{1+x^{10}}dx=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{\sin ^3 (\pi x)}{1+x^{10}}dx+\int_{\frac{\pi}{2}}^{\pi}\frac{\sin ^3 (\pi x)}{1+x^{10}}dx\]
\[=0+\int_{\frac{\pi}{2}}^{\pi}\frac{\sin ^3 (\pi x)}{1+x^{10}}dx.\]
Hence
\[\int_{-\frac{\pi}{2}}^{\pi}\frac{\sin ^3 (\pi x)}{1+x^{10}}dx=\int_{\frac{\pi}{2}}^{\pi}\frac{\sin ^3 (\pi x)}{1+x^{10}}dx.\]
Note that
\[\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{\sin ^3 (\pi x)}{1+x^{10}}dx=0\]
since the fuction $\frac{\sin ^3 (\pi x)}{1+x^{10}}$ is odd.
![Erdos](https://matchmaticians.com/storage/user/100028/thumb/matchmaticians-3empnt-file-5-avatar-512.jpg)
4.7K
Join Matchmaticians Affiliate Marketing
Program to earn up to a 50% commission on every question that your affiliated users ask or answer.
- 1 Answer
- 210 views
- Pro Bono
Related Questions
- What is this question asking and how do you solve it?
- Find the domain of the function $f(x)=\frac{\ln (1-\sqrt{x})}{x^2-1}$
- Show that $\psi:L(E,L(E,F))\rightarrow L^2(E,F)$ given by $[\psi(T)](u,v)=[T(u)](v)$ is a linear homeomorphism
- Mechanical principle help (maths)
- Find the absolute extrema of $f(x,y) = x^2 - xy + y^2$ on $|x| + |y| \leq 1$.
- Prove that $\lim_{n\rightarrow \infty} \int_{[0,1]^n}\frac{|x|}{\sqrt{n}}=\frac{1}{\sqrt{3}}$
- A question about the mathematical constant e.
- Finding absolute and relative extrema given an equation.
It is better to type body for Pro Bono questions and do not use links or images
I edited the question.