I-pi-6-pi-3-sin-2021-x-sin-2021-x-cos-2021-x-dx-

Question Number 144282 by SOMEDAVONG last updated on 24/Jun/21

I = \int_{\frac{π}{6}}^{\frac{π}{3}} \frac{\sin^{2021} x}{\sin^{2021} x + \cos^{2021} x} dx = ?

Answered by som(math1967) last updated on 24/Jun/21

I = \int_{\frac{π}{6}}^{\frac{π}{3}} \frac{{s i n}^{2021} (\frac{π}{3} + \frac{π}{6} - x)}{{s i n}^{2021} (\frac{π}{3} + \frac{π}{6} - x) + c o s (\frac{π}{3} + \frac{π}{6} - x)} d x

I = \int_{\frac{π}{6}}^{\frac{π}{3}} \frac{{c o s}^{2021} x}{\sin^{2021} x + \cos^{2021} x} dx

∴ 2 I = \int_{\frac{π}{6}}^{\frac{π}{3}} \frac{{c o s}^{2021} x + {s i n}^{2021} x}{\sin^{2021} x + \cos^{2021} x} dx

2 I = \int_{\frac{π}{6}}^{\frac{π}{3}} d x

I = {[\frac{x}{2}]}_{\frac{π}{6}}^{\frac{π}{3}} = \frac{π}{12} a n s

Commented by SOMEDAVONG last updated on 24/Jun/21

Thanks sir!