# 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

1

## 1 Answer

\[\int_{-\frac{\pi}{2}}^{\pi}\frac{\sin ^3 (\pi x)}{1+x^{10}}dx=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{\sin ^3 (\pi x)}{1+x^{10}}dx+\int_{\frac{\pi}{2}}^{\pi}\frac{\sin ^3 (\pi x)}{1+x^{10}}dx\]

\[=0+\int_{\frac{\pi}{2}}^{\pi}\frac{\sin ^3 (\pi x)}{1+x^{10}}dx.\]

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

4.6K

Join Matchmaticians Affiliate Marketing
Program to earn up to 50% commission on every question your affiliated users ask or answer.

- 1 Answer
- 175 views
- Pro Bono

### Related Questions

- Reduction formulae
- Two short calculus questions - domain and limits
- Find the limit as x --> +inf
- Evaluate $\int \sin x \sqrt{1+\cos x} dx$
- You have 100 feet of cardboard. You need to make a box with a square bottom, 4 sides, but no top.
- Please help me with this math problem I am struggling!
- Integral of trig functions
- Evaluate $\int_C (2x^3-y^3)dx+(x^3+y^3)dy$, where $C$ is the unit circle.

It is better to type body for Pro Bono questions and do not use links or images

I edited the question.