The integral we want to solve is:

\[
\int \frac{64}{x^2 + 64} \, dx
\]

### Step-by-step solution:

1. **Factor the denominator**: The denominator \(x^2 + 64\) can be rewritten as \(x^2 + 8^2\).

2. **Use the standard integral form**: The integral \(\int \frac{dx}{x^2 + a^2} = \frac{1}{a} \tan^{-1} \left( \frac{x}{a} \right) + C\), where \(a = 8\) in our case.

3. **Apply the constant factor**: We can pull out the constant \(64\) from the numerator and solve the integral of \(\frac{1}{x^2 + 64}\).

\[
\int \frac{64}{x^2 + 64} \, dx = 64 \int \frac{1}{x^2 + 8^2} \, dx
\]

4. **Apply the formula**:
\[
64 \int \frac{1}{x^2 + 8^2} \, dx = 64 \cdot \frac{1}{8} \tan^{-1} \left( \frac{x}{8} \right) + C
\]

5. **Simplify the constant**:
\[
= 8 \tan^{-1} \left( \frac{x}{8} \right) + C
\]

### Final answer:
\[
\int \frac{64}{x^2 + 64} \, dx = 8 \tan^{-1} \left( \frac{x}{8} \right) + C
\]

Would you like more details, or do you have any questions?

Here are 5 related questions to expand your understanding:
1. What is the geometric interpretation of the arctangent function in integrals?
2. How does changing the constant 64 in the numerator affect the solution?
3. Can you generalize the result for any \(x^2 + a^2\) in the denominator?
4. How can this integral be applied to solve problems involving trigonometric substitutions?
5. What is the role of inverse trigonometric functions in integration?

**Tip**: When you encounter integrals of the form \(\frac{1}{x^2 + a^2}\), it's useful to remember the result involves an arctangent function.

Solve the integral ∫ 64/(x^2 + 64) dx with a detailed step-by-step guide using inverse trigonometric functions. Learn about applying the arctangent formula for integrals of the form ∫ 1/(x^2 + a^2) dx. Suitable for undergraduate 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