intsin^(2)xcos^(2)xdx

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Recognize the integral: Recognize the integral that needs to be solved. We need to integrate the function sin^2(x)cos^2(x) with respect to x.Use trigonometric identities: Use trigonometric identities to simplify the integrand. We can use the double angle identity for cosine, which is cos(2x) = 2cos^2(x) - 1 or cos^2(x) = (1 + cos(2x))/2. Similarly, for sine, we have sin^2(x) = (1 - cos(2x))/2.Substitute the identities: Substitute the identities into the integral. The integral becomes ∫[(1 - cos(2x))/2] * [(1 + cos(2x))/2] dx.Expand the integrand: Expand the integrand. Expanding the integrand gives us ∫(1/4 - cos^2(2x)/4) dx.Split the integral: Split the integral into two separate integrals. We now have (1/4)∫dx - (1/4)∫cos^2(2x) dx.Integrate the first part: Integrate the first part of the integral. The integral of dx is x, so the first part becomes (1/4)x.Use a trigonometric identity: Use a trigonometric identity to simplify the second integral. We can use the power reduction identity for cos^2(2x), which is cos^2(2x) = (1 + cos(4x))/2.Substitute the identity: Substitute the identity into the second integral. The second integral becomes (1/4)∫(1/2 + cos(4x)/2) dx.Simplify the second integral: Simplify the second integral. This simplifies to (1/8)∫dx + (1/8)∫cos(4x) dx.Integrate the simplified second integral: Integrate the simplified second integral. The integral of dx is x, and the integral of cos(4x) is (1/4)sin(4x), so the second part becomes (1/8)x + (1/32)sin(4x).Combine the results: Combine the results from Step 6 and Step 10. The final answer is (1/4)x - [(1/8)x + (1/32)sin(4x)] + C, where C is the constant of integration.Simplify the final expression: Simplify the final expression. Combine like terms to get the final answer: (1/8)x - (1/32)sin(4x) + C.

$\int \sin^{2}x \cos^{2}x \, dx$

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∫sin⁡2xcos⁡2x dx\int \sin^{2}x \cos^{2}x \, dx∫sin2xcos2xdx

$\int \sin^{2}x \cos^{2}x \, dx$