find-pi-3-pi-3-x-2-cosx-sinx-3-dx-

Question Number 67235 by prof Abdo imad last updated on 24/Aug/19

f i n d \int_{- \frac{π}{3}}^{\frac{π}{3}} x^{2} {c o s x - s i n x}^{3} d x

Commented by mathmax by abdo last updated on 01/Sep/19

l e t I = \int_{- \frac{π}{3}}^{\frac{π}{3}} x^{2} {c o s x - s i n x}^{3} a n d J = \int_{- \frac{π}{6}}^{\frac{π}{6}} x^{2} {c o s x + s i n x}^{3} d x

w e h a v e I + J = \int_{- \frac{π}{3}}^{\frac{π}{3}} x^{2} (c o s x - s i n x + c o s x + s i n x) ({(c o s x - s i n x)}^{2}

- (c o s x - s i n x) (c o s x + s i n x) + {(c o s x + s i n x)}^{2}} d x

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

= \int_{- \frac{π}{3}}^{\frac{π}{3}} 2 x^{2} c o s x {2 - c o s (2 x)} d x

= 4 \int_{0}^{\frac{π}{3}} x^{2} c o s x (2 - c o s (2 x)) d x = 8 \int_{0}^{\frac{π}{3}} x^{2} c o s x d x - 4 \int_{0}^{\frac{π}{3}} c o s x . c o s (2 x) d x

b y p a r t s \int_{0}^{\frac{π}{3}} x^{2} c o s x d x = {[x^{2} s i n x]}_{0}^{\frac{π}{3}} - \int_{0}^{\frac{π}{3}} 2 x s i n x d x

= - 2 {{[- x c o s x]}_{0}^{\frac{π}{3}} - \int_{0}^{\frac{π}{3}} (- c o s x) d x}

= - 2 {- \frac{π}{6} + {[s i n x]}_{0}^{\frac{π}{3}}} = - 2 {- \frac{π}{6} + \frac{\sqrt{3}}{2}} = \frac{π}{3} - \sqrt{3}

\int_{0}^{\frac{π}{3}} c o s x c o s (2 x) d x = \frac{1}{2} \int_{0}^{\frac{π}{3}} (c o s (3 x) + c o s x) d x = \frac{1}{6} {[s i n (3 x)]}_{0}^{\frac{π}{3}}

+ \frac{1}{2} {[s i n x]}_{0}^{\frac{π}{3}} = \frac{1}{2} \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{4} \Rightarrow I + J = 8 (\frac{π}{3} - \sqrt{3}) - \sqrt{3} = \frac{8 π}{3} - 9 \sqrt{3}

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

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

= - 2 \int_{- \frac{π}{3}}^{\frac{π}{3}} x^{2} s i n x {2 + c o s (2 x)} d x

= 0 b e c a u s e t h e f u n c t i o n i s o d d \Rightarrow I - J = 0 \Rightarrow I = J \Rightarrow

\Rightarrow 2 I = \frac{8 π}{3} - 9 \sqrt{3} \Rightarrow I = \frac{4 π}{3} - \frac{9}{2} \sqrt{3}