I-pi-6-pi-3-cos-6-x-1-3sin-2-xcos-2-x-dx-

Question Number 143961 by SOMEDAVONG last updated on 20/Jun/21

I = \int_{\frac{π}{6}}^{\frac{π}{3}} \frac{\cos^{6} x}{1 - {3 \sin}^{2} {xcos}^{2} x} dx = ?

Answered by Dwaipayan Shikari last updated on 20/Jun/21

I = \int_{\frac{π}{6}}^{\frac{π}{3}} \frac{{c o s}^{6} x}{1 - 3 {s i n}^{2} x {c o s}^{2} x} d x \int_{ϵ}^{δ} f (δ + ϵ - x) d x = \int_{ϵ}^{δ} f (x) d x

= \int_{\frac{π}{6}}^{\frac{π}{3}} \frac{{s i n}^{6} x}{1 - 3 {s i n}^{2} {x c o s}^{2} x} = I 1 - 3 {s i n}^{2} x {c o s}^{2} x = {s i n}^{6} x + {c o s}^{6} x

2 I = \int_{\frac{π}{6}}^{\frac{π}{3}} \frac{{c o s}^{6} x + {s i n}^{6} x}{1 - 3 {s i n}^{2} {x c o s}^{2} x} d x = \int_{\frac{π}{6}}^{\frac{π}{3}} 1 d x = \frac{π}{6}

I = \frac{π}{12}

Commented by SOMEDAVONG last updated on 21/Jun/21

Thanks sir!