Question

Utilize the table of integrals to evaluate the following definite integral: ∫ 7 sin^6(8x) dx. Express your answer using 'c' for the constant of integration.

Answer 1

To evaluate the definite integral ∫ 7 sin^6(8x) dx, use the table of integrals and apply a trigonometric identity to simplify the function. Then integrate the simplified function to obtain the result.

Step-by-step explanation:

To evaluate the definite integral ∫ 7 sin6(8x) dx, we can use the table of integrals.

The integral can be expressed as:

∫ sin6(8x) dx

Using the trigonometric identity sin2(x) = (1 - cos(2x))/2, we can reduce the power of sin from 6 to 2.

Substituting this identity and simplifying, we have:

∫ (7/32) * (1 - cos(16x)) dx

Integrating, we get:

= (7/32) * (x - (1/16) * sin(16x)) + c

Therefore, the evaluated definite integral is (7/32) * (x - (1/16) * sin(16x)) + c.