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The integral of cos^2(x)sin^4(x) is (3x/8) - (1/32)sin(4x) + C.

To solve this integral, we can use the trigonometric identity cos^2(x) = (1 + cos(2x))/2 and sin^4(x) = (1 - cos(2x))^2/8. Substituting these identities into the integral, we get:

∫cos^2(x)sin^4(x) dx = ∫(1 + cos(2x))/2 * (1 - cos(2x))^2/8 dx

Expanding the numerator and simplifying, we get:

∫(1/16) - (3/16)cos(2x) + (1/8)cos^2(2x) dx

Using the identity cos^2(2x) = (1 + cos(4x))/2, we can simplify further:

∫(1/16) - (3/16)cos(2x) + (1/16) + (1/16)cos(4x) dx

Simplifying again, we get:

∫(3/32) - (3/16)cos(2x) + (1/16)cos(4x) dx

Integrating each term, we get:

(3x/32) - (3/32)sin(2x) + (1/64)sin(4x) + C

Simplifying, we get:

(3x/8) - (1/32)sin(4x) + C

Therefore, the integral of cos^2(x)sin^4(x) is (3x/8) - (1/32)sin(4x) + C. To fully understand this process, reviewing the `trigonometric identities`

involved can provide deeper insights. Furthermore, for those looking to refine their skills in solving such integrals, exploring various `techniques of integration`

is highly recommended. For specific applications involving trigonometric functions, the `integration of trigonometric functions`

can be particularly useful.

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