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
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
4.8K
Join Matchmaticians Affiliate Marketing
Program to earn up to a 50% commission on every question that your affiliated users ask or answer.
- 1 Answer
- 355 views
- Pro Bono
Related Questions
- A telephone line hanging between two poles.
- Is the infinite series $\sum_{n=1}^{\infty}\frac{1}{n \ln n}$ convergent or divergent?
- I need help with the attched problem about definite integrals
- Calculate $\iint_R (x+y)^2 e^{x-y}dx dy$ on the given region
- Find equation of the tangent line using implicit differentiation
- Application of integrals
- Riemann Sums for computing $\int_0^3 x^3 dx$
-
Limit graphs
It is better to type body for Pro Bono questions and do not use links or images
I edited the question.