Evaluate 

Integral from nothing to nothing cosine cubed StartFraction x Over 13 EndFraction dx∫cos3x13 dx.

To evaluate the integral of \(\cos^3\left(\frac{x}{13}\right)\) with respect to \(x\), we can use a trigonometric identity to simplify the integral. Specifically, we use the identity:

\[
\cos^3(\theta) = \cos(\theta) \cdot \cos^2(\theta) = \cos(\theta) \cdot \left(\frac{1 + \cos(2\theta)}{2}\right)
\]

Let \(\theta = \frac{x}{13}\). Then:

\[
\cos^3\left(\frac{x}{13}\right) = \cos\left(\frac{x}{13}\right) \cdot \frac{1 + \cos\left(\frac{2x}{13}\right)}{2}
\]

This expands to:

\[
\frac{1}{2}\cos\left(\frac{x}{13}\right) + \frac{1}{2}\cos\left(\frac{x}{13}\right)\cos\left(\frac{2x}{13}\right)
\]

So the integral becomes:

\[
\int \cos^3\left(\frac{x}{13}\right) \, dx = \frac{1}{2} \int \cos\left(\frac{x}{13}\right) \, dx + \frac{1}{2} \int \cos\left(\frac{x}{13}\right)\cos\left(\frac{2x}{13}\right) \, dx
\]

Now, we handle each integral separately.

### First Integral
The first term is straightforward:

\[
\frac{1}{2} \int \cos\left(\frac{x}{13}\right) \, dx
\]

The integral of \(\cos\left(\frac{x}{13}\right)\) is:

\[
13 \sin\left(\frac{x}{13}\right)
\]

So the first integral becomes:

\[
\frac{13}{2} \sin\left(\frac{x}{13}\right)
\]

### Second Integral
For the second integral:

\[
\frac{1}{2} \int \cos\left(\frac{x}{13}\right)\cos\left(\frac{2x}{13}\right) \, dx
\]

We use the product-to-sum identity:

\[
\cos(A)\cos(B) = \frac{1}{2}[\cos(A-B) + \cos(A+B)]
\]

So:

\[
\cos\left(\frac{x}{13}\right)\cos\left(\frac{2x}{13}\right) = \frac{1}{2}\left[\cos\left(-\frac{x}{13}\right) + \cos\left(\frac{3x}{13}\right)\right]
\]

Since \(\cos(-\theta) = \cos(\theta)\), we simplify to:

\[
\frac{1}{2}\left[\cos\left(\frac{x}{13}\right) + \cos\left(\frac{3x}{13}\right)\right]
\]

So the second integral becomes:

\[
\frac{1}{4} \int \cos\left(\frac{x}{13}\right) \, dx + \frac{1}{4} \int \cos\left(\frac{3x}{13}\right) \, dx
\]

The integrals are:

\[
\frac{13}{4} \sin\left(\frac{x}{13}\right) + \frac{13}{12} \sin\left(\frac{3x}{13}\right)
\]

### Final Answer
Combining everything:

\[
\int \cos^3\left(\frac{x}{13}\right) \, dx = \frac{13}{2} \sin\left(\frac{x}{13}\right) + \frac{13}{8} \sin\left(\frac{x}{13}\right) + \frac{13}{12} \sin\left(\frac{3x}{13}\right) + C
\]

Simplifying:

\[
\int \cos^3\left(\frac{x}{13}\right) \, dx = \frac{39}{8} \sin\left(\frac{x}{13}\right) + \frac{13}{12} \sin\left(\frac{3x}{13}\right) + C
\]

where \(C\) is the constant of integration.

Do you have any questions or need further clarification on this?

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Here are 5 related questions to consider:
1. What is the integral of \(\cos^2\left(\frac{x}{13}\right)\) with respect to \(x\)?
2. Can you integrate \(\cos^4\left(\frac{x}{13}\right)\) using similar identities?
3. How does the process differ when integrating \(\sin^3\left(\frac{x}{13}\right)\)?
4. What is the indefinite integral of \(\cos^2(x)\sin(x)\) with respect to \(x\)?
5. How do you apply integration by parts to trigonometric integrals?

**Tip:** Utilizing trigonometric identities can greatly simplify integrals involving powers of trigonometric functions.

Learn how to evaluate the integral of cos^3(x/13) step-by-step. This problem involves using trigonometric identities and calculus techniques. Suitable for advanced undergraduate students and anyone interested in integral calculus and trigonometry.

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