# What is #int cos^2(4x)dx #?

using the formulas for double angles.

Reexpressing the integral thus provides us with:

And right now, integrating is simple and yields:

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To integrate ( \cos^2(4x) ) with respect to ( x ), we can use the trigonometric identity ( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} ). Thus, we have:

[ \int \cos^2(4x) , dx = \int \frac{1 + \cos(8x)}{2} , dx ]

Now, we integrate each term separately:

[ \int \frac{1}{2} , dx = \frac{1}{2}x + C ]

[ \int \frac{\cos(8x)}{2} , dx = \frac{1}{16} \sin(8x) + C ]

Therefore,

[ \int \cos^2(4x) , dx = \frac{1}{2}x + \frac{1}{16} \sin(8x) + 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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