How to evaluate $\int\sin ^3 x\cos^3 x\:dx$ without a reduction formula?

Question

We have the integral $$\displaystyle\int \sin ^3 x \cos^3 x \:dx.$$ You can do this using the reduction formula, but I wonder if there's another (perhaps simpler) way to do this, like for example with a substitution?

Answer 1

Hint. You may write $$ \sin^3 x \cos^3 x= \sin x(1 - \cos^2x)\cos^3 x=\sin x(\cos^3x - \cos^5x) $$

Answer 2

You may write $$\sin^3 x \cos^3 x= \frac{1}{8} \sin^3(2 x)= \frac{1}{32}(3 \sin 2x - \sin 6x)$$ that is easy to integrate.

Answer 3

Multiple arguments subscripted shorthand:

$$ 8 s^3 c^3 = s_2^3 = \frac{3 s_2 - s_6 }{4} $$

Normal notation:

$$ 8 \sin ^3 x \cos ^3 x = \sin ^3 2x = \dfrac {3 \sin 2x - \sin 6x }{4} $$

Integrate.

$$ \frac{1}{32}(3 \sin 2x - \sin 6x)$$ that easily integrates to

$$ \frac{1}{32}\big( \frac{-3 \cos 2x}{2} +\frac{ \cos 6x)}{6}\big) + C $$