# How do you find the integral of #cos^5(3x)dx#?

Now it's a simple polynomial integral we can evaluate it straightforwardly

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To find the integral of cos^5(3x)dx, you can use the method of substitution. Let u = 3x, then du = 3dx. Rearranging gives dx = du/3.

So, the integral becomes ∫cos^5(u)(du/3).

Now, rewrite cos^5(u) as (cos^2(u))^2 * cos(u), then substitute cos^2(u) = 1 - sin^2(u).

The integral becomes ∫(1 - sin^2(u))^2 * cos(u) * (du/3).

Expand (1 - sin^2(u))^2, which simplifies to 1 - 2sin^2(u) + sin^4(u).

Now the integral is ∫(1 - 2sin^2(u) + sin^4(u)) * cos(u) * (du/3).

Distribute and integrate each term separately.

∫(cos(u) - 2sin^2(u)cos(u) + sin^4(u)cos(u)) * (du/3).

Now, integrate each term:

∫cos(u) * (du/3) - 2∫sin^2(u)cos(u) * (du/3) + ∫sin^4(u)cos(u) * (du/3).

The integral of cos(u) is sin(u), the integral of sin^2(u)cos(u) is -(1/3)sin^3(u), and the integral of sin^4(u)cos(u) is (1/5)sin^5(u).

So, the final answer is (1/3)sin(u) - (2/3)sin^3(u) + (1/15)sin^5(u) + C, where C is the constant of integration.

Substitute back u = 3x to get the final answer in terms of x:

(1/3)sin(3x) - (2/3)sin^3(3x) + (1/15)sin^5(3x) + C.

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To find the integral of (\cos^5(3x) , dx), we can use the method of integration by parts or trigonometric identities. One approach is to use the half-angle identity for cosine, which states that (\cos^2(x) = \frac{1 + \cos(2x)}{2}). By repeatedly applying this identity, we can express (\cos^5(3x)) in terms of cosines of multiples of (3x). Then, we can integrate each term individually.

Another approach is to use a reduction formula for powers of cosine. However, this method can become complex for higher powers of cosine.

Ultimately, the choice of method depends on personal preference and familiarity with integration techniques. Both methods can be used to find the integral of (\cos^5(3x) , dx).

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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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