Question

Evaluate the integral ∫(3sin³(x)cos(x))dx. Remember to use absolute values where appropriate. Use c for the constant of integration.

Answer 1

To evaluate the integral ∫(3sin³(x)cos(x))dx, we can use the power reduction formula for sin³(x). Applying this formula, we can split the integral into two separate integrals and apply the power rule for integration. Integrating each term separately and simplifying using the double-angle formula for sin(2x), we obtain the final result.

Step-by-step explanation:

To evaluate the integral ∫(3sin³(x)cos(x))dx, we can use the power reduction formula for sin³(x). The formula states that sin³(x) = (3/4)sin(x) - (1/4)sin(3x). Applying this formula, we get:

∫(3sin³(x)cos(x))dx = ∫((3/4)sin(x)cos(x) - (1/4)sin(3x)cos(x))dx

Now, we can split the integral into two separate integrals and apply the power rule for integration:

∫((3/4)sin(x)cos(x))dx - ∫((1/4)sin(3x)cos(x))dx

Integrating each term separately, we obtain:

(3/4)∫(sin(x)cos(x))dx - (1/4)∫(sin(3x)cos(x))dx

Using the double-angle formula for sin(2x), we can simplify further:

(3/4)∫(sin(2x)/2)dx - (1/4)∫(sin(4x)/2)dx

Finally, integrating each term, we have:

(3/8)cos(2x) - (1/8)cos(4x) + C