# How do you find the antiderivative of # cos^4 (x) dx#?

You want to split it up using trig identities to get nice, easy integrals.

So,

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To find the antiderivative of cos^4(x) dx, you can use the reduction formula for powers of cosine. The reduction formula states that:

∫cos^n(x) dx = (cos^(n-1)(x) * sin(x))/n + (n-1)/n * ∫cos^(n-2)(x) dx

Using this formula, you can find the antiderivative of cos^4(x) dx. Start by setting n = 4:

∫cos^4(x) dx = (cos^3(x) * sin(x))/4 + (3/4) * ∫cos^2(x) dx

Now, you need to simplify ∫cos^2(x) dx using the reduction formula again. Set n = 2:

∫cos^2(x) dx = (cos(x) * sin(x))/2 + (1/2) * ∫dx ∫cos^2(x) dx = (cos(x) * sin(x))/2 + (1/2) * x + C

Now, substitute this result back into the first equation:

∫cos^4(x) dx = (cos^3(x) * sin(x))/4 + (3/4) * ((cos(x) * sin(x))/2 + (1/2) * x) + C ∫cos^4(x) dx = (cos^3(x) * sin(x))/4 + (3/8) * cos(x) * sin(x) + (3/8) * x + C

So, the antiderivative of cos^4(x) dx is (cos^3(x) * sin(x))/4 + (3/8) * cos(x) * sin(x) + (3/8) * 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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