Find the indefinite integral by making a change of variables. (Remember the constant of integration.)

cos7(8*x*) sin(8*x*) *dx*

To solve this indefinite integral by substitution, let's proceed step-by-step:

### Problem
\[
\int \cos^7(8x) \sin(8x) \, dx
\]

### Step 1: Choose a Substitution
Let's set:
\[
u = \cos(8x)
\]
Then, the derivative of \( u \) with respect to \( x \) is:
\[
\frac{du}{dx} = -8 \sin(8x)
\]
or rearranging,
\[
\sin(8x) \, dx = -\frac{1}{8} \, du
\]

### Step 2: Substitute in Terms of \( u \)
Rewrite the integral using \( u \):
\[
\int \cos^7(8x) \sin(8x) \, dx = \int u^7 \cdot \left(-\frac{1}{8}\right) \, du
\]
This simplifies to:
\[
-\frac{1}{8} \int u^7 \, du
\]

### Step 3: Integrate with Respect to \( u \)
Now integrate \( u^7 \) with respect to \( u \):
\[
-\frac{1}{8} \cdot \frac{u^8}{8} = -\frac{1}{64} u^8 + C
\]

### Step 4: Substitute Back in Terms of \( x \)
Since \( u = \cos(8x) \), substitute back:
\[
-\frac{1}{64} \cos^8(8x) + C
\]

### Final Answer
\[
\int \cos^7(8x) \sin(8x) \, dx = -\frac{1}{64} \cos^8(8x) + C
\]

Would you like further clarification on any of the steps? 

---

Here are some related questions to consider:

1. How would the solution change if the power of cosine were different?
2. What if the integral involved \( \cos^7(8x) \cos(8x) \) instead?
3. Can you solve similar integrals using trigonometric identities instead of substitution?
4. How does the constant \( C \) affect the family of solutions?
5. What techniques can be used for integrals without obvious substitutions?

**Tip:** Always double-check substitutions by differentiating back to verify if the solution matches the original integrand.

Find the indefinite integral by making a change of variables. (Remember the constant of integration.)

cos^7(8x) * sin(8x) dx

This guide provides a detailed solution for finding the indefinite integral of cos^7(8x) * sin(8x) dx using the substitution method. Key concepts include trigonometric integration and substitution techniques, making it ideal for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation