# How do you find the antiderivative of #sin^2 (x)cos^2 (x) dx#?

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To find the antiderivative of sin^2(x)cos^2(x) dx, you can use the double angle formula for sine and cosine, which states that sin(2θ) = 2sin(θ)cos(θ) and cos(2θ) = cos^2(θ) - sin^2(θ). Then, use a substitution to simplify the integral. Here's the step-by-step process:

- Use the double angle formula for sine: sin^2(x) = (1 - cos(2x)) / 2.
- Use the double angle formula for cosine: cos^2(x) = (1 + cos(2x)) / 2.
- Substitute these into the integral: sin^2(x)cos^2(x) = ((1 - cos(2x)) / 2) * ((1 + cos(2x)) / 2).
- Simplify: sin^2(x)cos^2(x) = (1 - cos^2(2x)) / 4.
- Use another trigonometric identity: cos^2(2x) = (1 + cos(4x)) / 2.
- Substitute: sin^2(x)cos^2(x) = (1 - (1 + cos(4x)) / 2) / 4 = (cos(4x) - 1) / 8.
- Integrate: ∫(sin^2(x)cos^2(x)) dx = ∫((cos(4x) - 1) / 8) dx = (1/8) ∫(cos(4x) - 1) dx.
- Integrate each term: (1/8) ∫(cos(4x) - 1) dx = (1/8) * ((1/4)sin(4x) - x) + C.
- Simplify: (1/32)sin(4x) - (1/8)x + C.

Therefore, the antiderivative of sin^2(x)cos^2(x) dx is (1/32)sin(4x) - (1/8)x + C, where C is the constant of integration.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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