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The integral presented is:

\[
\int 19 \cos^3(x) \, dx
\]

The first step suggests using a trigonometric identity to simplify \(\cos^3(x)\), specifically the Pythagorean identity \(\sin^2(x) + \cos^2(x) = 1\). To proceed, we express \(\cos^3(x)\) as a product of \(\cos(x)\) and \(\cos^2(x)\), and then replace \(\cos^2(x)\) with \(1 - \sin^2(x)\):

\[
\cos^3(x) = \cos(x) \cdot \cos^2(x) = \cos(x)(1 - \sin^2(x))
\]

Now, the integral becomes:

\[
\int 19 \cos(x)(1 - \sin^2(x)) \, dx
\]

At this point, we can use substitution, letting \(u = \sin(x)\), so that \(du = \cos(x) \, dx\). Substituting these into the integral:

\[
\int 19 (1 - u^2) \, du
\]

This integral can be split into two simpler integrals:

\[
19 \int (1 - u^2) \, du = 19 \left( \int 1 \, du - \int u^2 \, du \right)
\]

Now, integrate each term:

\[
19 \left( u - \frac{u^3}{3} \right) + C
\]

Finally, substitute \(u = \sin(x)\) back:

\[
19 \left( \sin(x) - \frac{\sin^3(x)}{3} \right) + C
\]

This is the solution to the given integral.

Would you like further clarification, or any specific steps explained in more detail?

Here are 5 related questions you might find useful:
1. How do you apply substitution for trigonometric integrals in general?
2. What are other strategies for integrating odd powers of trigonometric functions?
3. How can we apply integration techniques to \(\sin^n(x)\) when \(n\) is odd or even?
4. How does substitution work when using other trigonometric identities?
5. Can integration by parts be used for similar integrals?

**Tip**: When integrating powers of sine and cosine, always check for simplifications using identities like \(\sin^2(x) + \cos^2(x) = 1\).

Learn how to evaluate the integral of 19 cos^3(x) with a step-by-step guide using trigonometric identities and substitution. This solution covers key calculus concepts including the use of Pythagorean identities and substitution. Ideal for advanced high school or undergraduate students.

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