# Evaluate the integral? : #int 8cos^4 2pit dt #

# int \ 8cos^4 2pit \ dt = 3x + (sin(4pix))/(pi)+(sin(8pix))/(8pi) + C #

We seek:

We can utilise the double angle identity:

From which we get:

And squaring we get:

To make thing easier to read we can also perform a simple substitution:

So our integral becomes:

And restraining the substitution we have:

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To evaluate the integral ∫8cos^4(2πt) dt:

- Use the identity cos^2(θ) = (1 + cos(2θ))/2.
- Replace cos^4(2πt) with (1 + cos(4πt))/2.
- Distribute the 8: 8/2 = 4.
- Integrate term by term: ∫4(1 + cos(4πt))/2 dt.
- Integrate each term separately: 4/2 * ∫(1 + cos(4πt)) dt.
- Integrate each term: 4/2 * [t + (1/(4π))sin(4πt)] + C.
- Simplify: 2t + (2/(π))sin(4πt) + C.

So, the integral of 8cos^4(2πt) dt equals 2t + (2/(π))sin(4πt) + C.

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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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