#Integrating Composite Functions

#Table of Contents

1. Introduction to Integrating Composite Functions
2. Steps for Integrating Composite Functions
3. Special Case: Function Raised to a Power
4. Integrating $f'(x)/f(x)$
5. Practice Questions
6. Glossary
7. Summary and Key Takeaways

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#Introduction to Integrating Composite Functions

#What is Meant by Integrating Composite Functions?

Integrating composite functions refers to the process of integrating by inspection, often using the reverse chain rule. This method involves recognizing that the chain rule used in differentiation can be reversed for integration.

In function notation, this method is used to integrate expressions of the form:

$$\int f'(g(x)) \cdot g'(x) , dx$$

By the chain rule:

$$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$$

Since differentiation and integration are inverse operations:

$$\int f'(g(x)) \cdot g'(x) , dx = f(g(x)) + C$$

#Example of Adjust and Compensate

Consider the integral:

$$5x^2 , dx$$

Here, 5x^2 is not quite the derivative of $g(x) = 4x^3$ because:

$$g'(x) = 12x^2$$

To adjust and compensate, we rewrite:

$$5x^2 = \frac{5}{12} (12x^2) = \frac{5}{12} g'(x)$$

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#Steps for Integrating Composite Functions

#Step 1: Identify the Main Function

Spot the main function in the integral. For example...