∫ cos^3 (2x) sin^3 (3x) dx


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Question Number 91961 by jagoll last updated on 04/May/20

$\int \cos^{3} (2 x) \sin^{3} (3 x) dx$
Commented by mathmax by abdo last updated on 04/May/20
$cos(2x)sin(3x) =cos(2x)cos((π/2)−3x) =(1/2){cos(−x+(π/2))+cos(5x−(π/2))} =(1/2){ sinx +sin(5x)} ⇒ cos^3 (2x)sin^3 (3x) =(1/8){sinx +sin(5x)}^3 =(1/8){ sin^3 x +3sin^2 x sin(5x)+3sinxsin^2 (5x) +sin^3 (5x)} sin^3 x =sin^2 x sinx =((1−cos(2x))/2)sinx =(1/2)sinx −(1/2) cos(2x)sinx but cos(2x)sinx =cos(2x)cos((π/2)−x) =(1/2){ cos(x+(π/2))+cos(3x−(π/2))} =(1/2){−sinx +sin(3x)} ⇒ sin^3 x =(1/2)sin(x)−(1/4)(−sinx +sin(3x)) =(3/4)sinx−(1/4)sin(3x) ⇒ 8I = ∫ ((3/4)sinx−(1/4)sin(3x))dx +3∫ sin^2 x sin(5x)dx +3 ∫ sinx sin^2 5x +∫ ((3/4)sin(5x) −(1/4)sin(15x))dx also sin^2 x sin(5x) =(1/2)(1−cos(2x))sin(5x) =(1/2)sin(5x)−(1/2)cos(2x)sin(5x) but cos(2x)sin(5x) =cos(2x)cos((π/2)−5x) =(1/2)(cos(−3x+(π/2))+cos(7x−(π/2))) =(1/2)(sin(3x)+sin(7x)) now its eazy to solve the integral...$
$c o s (2 x) s i n (3 x) = c o s (2 x) c o s (\frac{π}{2} - 3 x)$ $= \frac{1}{2} {c o s (- x + \frac{π}{2}) + c o s (5 x - \frac{π}{2})}$ $= \frac{1}{2} {s i n x + s i n (5 x)} \Rightarrow$ ${c o s}^{3} (2 x) {s i n}^{3} (3 x) = \frac{1}{8} {s i n x + s i n (5 x)}^{3}$ $= \frac{1}{8} {{s i n}^{3} x + 3 {s i n}^{2} x s i n (5 x) + 3 {s i n x s i n}^{2} (5 x) + {s i n}^{3} (5 x)}$ ${s i n}^{3} x = {s i n}^{2} x s i n x = \frac{1 - c o s (2 x)}{2} s i n x$ $= \frac{1}{2} s i n x - \frac{1}{2} c o s (2 x) s i n x b u t c o s (2 x) s i n x = c o s (2 x) c o s (\frac{π}{2} - x)$ $= \frac{1}{2} {c o s (x + \frac{π}{2}) + c o s (3 x - \frac{π}{2})}$ $= \frac{1}{2} {- s i n x + s i n (3 x)} \Rightarrow$ ${s i n}^{3} x = \frac{1}{2} s i n (x) - \frac{1}{4} (- s i n x + s i n (3 x))$ $= \frac{3}{4} s i n x - \frac{1}{4} s i n (3 x) \Rightarrow$ $8 I = \int (\frac{3}{4} s i n x - \frac{1}{4} s i n (3 x)) d x + 3 \int {s i n}^{2} x s i n (5 x) d x$ $+ 3 \int s i n x {s i n}^{2} 5 x + \int (\frac{3}{4} s i n (5 x) - \frac{1}{4} s i n (15 x)) d x$ $a l s o {s i n}^{2} x s i n (5 x) = \frac{1}{2} (1 - c o s (2 x)) s i n (5 x)$ $= \frac{1}{2} s i n (5 x) - \frac{1}{2} c o s (2 x) s i n (5 x) b u t$ $c o s (2 x) s i n (5 x) = c o s (2 x) c o s (\frac{π}{2} - 5 x)$ $= \frac{1}{2} (c o s (- 3 x + \frac{π}{2}) + c o s (7 x - \frac{π}{2}))$ $= \frac{1}{2} (s i n (3 x) + s i n (7 x)) n o w i t s e a z y t o s o l v e t h e i n t e g r a l . . .$
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