https://file.aiquickdraw.com/mathbot/file/1726324884787w0w93mxk.jpg

Solve this problem showing all the steps.

This integral can be solved in steps to match the provided result. Here's a step-by-step approach to compute the integral:

\[
I = \int_0^{\frac{\pi}{6}} \cos^4(3\theta) \sin^2(6\theta) \, d\theta
\]

### Step 1: Simplify Using Trigonometric Identities
We will use two identities to simplify the powers of sine and cosine.

1. \( \cos^2(x) = \frac{1 + \cos(2x)}{2} \)
2. \( \sin^2(x) = \frac{1 - \cos(2x)}{2} \)

Now, apply these identities to both \( \cos^4(3\theta) \) and \( \sin^2(6\theta) \).

#### Simplifying \( \cos^4(3\theta) \):
\[
\cos^4(3\theta) = \left( \cos^2(3\theta) \right)^2 = \left( \frac{1 + \cos(6\theta)}{2} \right)^2
\]
\[
\cos^4(3\theta) = \frac{1}{4} \left( 1 + 2\cos(6\theta) + \cos^2(6\theta) \right)
\]
Now, simplify \( \cos^2(6\theta) \):
\[
\cos^2(6\theta) = \frac{1 + \cos(12\theta)}{2}
\]
So,
\[
\cos^4(3\theta) = \frac{1}{4} \left( 1 + 2\cos(6\theta) + \frac{1 + \cos(12\theta)}{2} \right)
\]
Simplifying this further:
\[
\cos^4(3\theta) = \frac{1}{4} \left( \frac{3}{2} + 2\cos(6\theta) + \frac{\cos(12\theta)}{2} \right)
\]

#### Simplifying \( \sin^2(6\theta) \):
\[
\sin^2(6\theta) = \frac{1 - \cos(12\theta)}{2}
\]

### Step 2: Expand the Product
Now, multiply \( \cos^4(3\theta) \) by \( \sin^2(6\theta) \):
\[
\cos^4(3\theta) \sin^2(6\theta) = \left( \frac{1}{4} \left( \frac{3}{2} + 2\cos(6\theta) + \frac{\cos(12\theta)}{2} \right) \right) \cdot \frac{1 - \cos(12\theta)}{2}
\]
This product will give several terms which can be integrated term by term.

### Step 3: Integrate Term by Term
Each term in the expansion can now be integrated over \( \theta \) from \( 0 \) to \( \frac{\pi}{6} \).

After performing this process (either by hand or through a more detailed symbolic approach

Show that \( \int_0^{\frac{\pi}{6}} \cos^4(3\theta) \sin^2(6\theta) \, d\theta = \frac{5\pi}{192} \).

This step-by-step solution demonstrates how to solve the trigonometric integral \( \int_0^{\frac{\pi}{6}} \cos^4(3\theta) \sin^2(6\theta) \, d\theta \) using trigonometric identities and integration techniques, arriving at the result \( \frac{5\pi}{192} \). Suitable for advanced calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation