Evaluate the indefinite integral.

integral of 8 sin^4 x cos x dx

To evaluate the indefinite integral of 8 sin^4(x) cos(x) dx, we can use a trigonometric identity to simplify the expression. The trigonometric identity we will use is the double angle identity for cosine, which states that:

cos(2x) = 2cos^2(x) - 1

We can rearrange this identity to solve for cos^2(x):

cos^2(x) = (1 + cos(2x)) / 2

Now, let's look at the original integral:

∫ 8 sin^4(x) cos(x) dx

We can rewrite sin^4(x) as (sin^2(x))^2:

∫ 8 (sin^2(x))^2 cos(x) dx

Next, we can use the substitution u = sin^2(x), du = 2sin(x) cos(x) dx:

∫ 8 (u^2) (du/2) = 4∫ u^2 du

Now, we can substitute back u as sin^2(x) and rewrite the integral:

= 4∫ sin^4(x) dx

Using the double angle identity for cosine, we can rewrite sin^2(x) as (1 - cos(2x))/2:

= 4∫ [(1 - cos(2x))/2]^2 dx

Expanding this expression:

= 4∫ (1 - 2cos(2x) + cos^2(2x))/4 dx

= ∫ (1 - 2cos(2x) + cos^2(2x)) dx

Now, we can integrate each term separately:

∫ 1 dx = x

∫ 2cos(2x) dx = sin(2x)

∫ cos^2(2x) dx can be evaluated by using a power reduction formula to convert cos^2(2x) into a sum of trigonometric functions.

The power reduction formula for cos^2(2x) is:

cos^2(2x) = (1 + cos(4x))/2

We can rewrite the integral as:

∫ (1 + cos(4x))/2 dx

∫ (1/2) dx + ∫ (cos(4x)/2) dx

= (1/2) x + (1/8) sin(4x)

Finally, putting all the terms together:

Integral of 8 sin^4(x) cos(x) dx = (4x + 2sin(2x) + (1/2) x + (1/8) sin(4x)) + C

where C is the constant of integration.