Question

Calcule.$\int\limits {sen 3x}\ sen5 \, xdx$

Answer 1

$\sin(a) \sin(b) \\ = \frac{1}{2} ( \cos(a - b) - \cos(a + b) )$

$sin(3x). \sin(5x) = \\ \frac{ ( \cos(3x -5x) - \cos(3x + 5x)) }{2} \\ = \frac{1}{2}( \cos( - 2x) - \cos(8x) )$

$∫( \sin(3x) \sin(5x)) dx = \\ \frac{1}{2} ∫ (\cos( - 2x) - \cos(8x) )dx$

$\frac{1}{2} ∫ \cos(2x) dx - \frac{1}{2}∫ \cos(8x)dx \\ = \frac{1}{4} \sin(2x) - \frac{1}{16} \sin(8x) + c$