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What's the integral of cos^2(x)sin^4(x)?

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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