Evaluate the following integral.
Integral from nothing to nothing 5 cosine cubed left parenthesis 4 x right parenthesis dx



The integral you're referring to is:

\[
\int 5 \cos^3(4x) \, dx
\]

To solve this, we'll use a trigonometric identity to simplify the expression.

1. Start with the identity for \(\cos^3(x)\):

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

Thus, the given integral becomes:

\[
\int 5 \cos^3(4x) \, dx = \int 5 \cos(4x) \left(\frac{1 + \cos(8x)}{2}\right) \, dx
\]

2. Simplify the integral:

\[
= \frac{5}{2} \int \cos(4x)(1 + \cos(8x)) \, dx
\]

Now split this into two integrals:

\[
= \frac{5}{2} \left( \int \cos(4x) \, dx + \int \cos(4x) \cos(8x) \, dx \right)
\]

3. Solve each integral separately:

For the first integral:
\[
\int \cos(4x) \, dx = \frac{1}{4} \sin(4x)
\]

For the second integral, we can use the product-to-sum identity:

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

Thus:

\[
\cos(4x) \cos(8x) = \frac{1}{2}[\cos(-4x) + \cos(12x)] = \frac{1}{2}[\cos(4x) + \cos(12x)]
\]

Now integrate both terms:

\[
\int \cos(4x) \cos(8x) \, dx = \frac{1}{2} \left( \int \cos(4x) \, dx + \int \cos(12x) \, dx \right)
\]

The integrals are straightforward:

\[
\int \cos(12x) \, dx = \frac{1}{12} \sin(12x)
\]

So now we combine everything:

\[
\frac{5}{2} \left( \frac{1}{4} \sin(4x) + \frac{1}{2} \left( \frac{1}{4} \sin(4x) + \frac{1}{12} \sin(12x) \right) \right)
\]

Simplify this expression:

\[
= \frac{5}{8} \sin(4x) + \frac{5}{24} \sin(12x) + C
\]

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

Thus, the final answer is:

\[
\boxed{\frac{5}{8} \sin(4x) + \frac{5}{24} \sin(12x) + C}
\]

---

Would you like a more detailed explanation of any step? Here are 5 related questions:

1. What is the product-to-sum identity for trigonometric functions?
2. How do you use substitution to solve integrals like \(\int \cos(4x) \, dx\)?
3. Why do we split the original integral using trigonometric identities?
4. What are common trigonometric identities used in integration?
5. How do you evaluate more complex trigonometric integrals?

**Tip:** Always check for trigonometric identities that can simplify powers of sine or cosine before integrating directly!

Evaluate the following integral. Integral from nothing to nothing 5 cosine cubed left parenthesis 4 x right parenthesis dx.

Learn how to solve the integral ∫ 5 cos^3(4x) dx using trigonometric identities and product-to-sum formulas. This step-by-step guide is suitable for college-level calculus and explains key techniques in integral calculus and trigonometric simplifications.

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