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Question Number 75093 by Rio Michael last updated on 07/Dec/19

\int {c o s}^{3} {x s i n}^{3} x d x

Commented by mathmax by abdo last updated on 07/Dec/19

I = \int {c o s}^{3} x {s i n}^{3} x d x = \int {(c o s x s i n x)}^{3} d x

= \frac{1}{8} \int {(s i n (2 x))}^{3} d x = \frac{1}{8} \int s i n (2 x) {s i n}^{2} (2 x) d x

= \frac{1}{8} \int s i n (2 x) (\frac{1 - c o s (4 x)}{2}) d x

= \frac{1}{16} \int s i n (2 x) d x - \frac{1}{16} \int s i n (2 x) c o s (4 x) d x

s i n (2 x) c o s (4 x) = c o s (\frac{π}{2} - 2 x) c o s (4 x)

= \frac{1}{2} {c o s (\frac{π}{2} + 2 x) + c o s (\frac{π}{2} - 6 x)} = \frac{1}{2} {s i n (6 x) - s i n (2 x)} \Rightarrow

I = - \frac{1}{32} c o s (2 x) - \frac{1}{32} \int (s i n (6 x) - s i n (2 x)) d x

= - \frac{1}{32} c o s (2 x) + \frac{1}{6 \times 32} c o s (6 x) - \frac{1}{64} c o s (2 x) + C

= - \frac{3}{64} c o s (2 x) + \frac{1}{192} c o s (6 x) + C

Answered by mr W last updated on 07/Dec/19

\int {c o s}^{3} {x s i n}^{3} x d x

= \int {c o s}^{2} {x s i n}^{3} x d (\sin x)

= \int (1 - \sin^{2} x) {s i n}^{3} x d (\sin x)

= \int (\sin^{3} x - \sin^{5} x) d (\sin x)

= \frac{\sin^{4} x}{4} - \frac{\sin^{6} x}{6} + C

Commented by Rio Michael last updated on 07/Dec/19

t h a n k y o u s i r, s o m u c h