Final answer:
To evaluate the integral ∫ 19 sin²(x) cos³(x) dx, we can use power reducing identities to simplify the integrand and then integrate each term separately.
Step-by-step explanation:
To evaluate the integral ∫ 19 sin²(x) cos³(x) dx, we can use the power reducing identities to simplify the integrand. The power reducing identities state that sin²(x) = (1 - cos(2x))/2 and cos³(x) = (1/4)(3cos(x) + cos(3x)). Substituting these identities into the integral gives us:
∫ 19(1 - cos(2x))/2 * (1/4)(3cos(x) + cos(3x)) dx
Expanding and simplifying this expression, we get:
∫ (19/8)(3cos(x) - 3cos(3x) - cos(3x) + cos(5x)) dx
Now, integrate each term separately:
(19/8)(∫ 3cos(x) dx - ∫ 3cos(3x) dx - ∫ cos(3x) dx + ∫ cos(5x) dx)
= (19/8)(3sin(x) - (3/3)sin(3x) - (1/3)sin(3x) + (1/5)sin(5x)) + C
where C is the constant of integration.