$$\int x(5x^2 - 2)^6 , dx$$

Step-by-step:

1. Identify the main function: $(5x^2 - 2)^6$
2. Adjust and compensate: $\frac{1}{70} \int 70x(5x^2 - 2)^6 , dx$
3. Integrate and simplify: $\frac{1}{70} (5x^2 - 2)^7 + C$

#Integrating $f'(x)/f(x)$

#How to Integrate $f'(x)/f(x)$?

A particularly useful special case of integrating composite functions is:

$$\int \frac{f'(x)}{f(x)} , dx = \ln |f(x)| + C$$

This pattern occurs when the numerator of a fraction being integrated is the derivative of the denominator.

#Example of Adjust and Compensate

Consider the integral:

$$\int \frac{x^2 + 1}{x^3 + 3x} , dx$$

Adjust and compensate:

$$\frac{1}{3} \int 3 \frac{x^2 + 1}{x^3 + 3x} , dx = \frac{1}{3} \int \frac{3x^2 + 3}{x^3 + 3x} , dx = \frac{1}{3} \ln |x^3 + 3x| + C$$

#Worked Example

#Example: Finding an Expression for $f(x)$

Let $f$ be a function whose derivative, $f'$, is given by:

$$f'(x) = 5x^2 \sin(2x^3)$$

Given that the graph of $f$ passes through the point $(0, 1)$, find an expression for $f(x)$.

Solution:

Use:

$$f(x) = \int f'(x) , dx$$

1. Identify the main function: $\sin(\dots)$
2. Recognize that $\sin$ comes from $\cos$, so use the chain rule:

$$\frac{d}{dx}[\cos(2x^3)] = -\sin(2x^3) \cdot 6x^2$$

3. Adjust and compensate:

$$f(x) = \int 5x^2 \sin(2x^3) , dx = -\frac{5}{6} \int (-\sin(2x^3) \cdot 6x^2) , dx$$

4. Integrate:

$$f(x) = -\frac{5}{6} \cos(2x^3) + C$$

Since the graph of $f$ goes through $(0, 1)$:

$$f(0) = 1$$

$$-\frac{5}{6} \cos(0) + C = 1$$

$$-\frac{5}{6} + C = 1$$

$$C = \frac{11}{6}$$

Therefore:

$$f(x) = -\frac{5}{6} \cos(2x^3) + \frac{11}{6}$$

#Practice Questions

#Glossary

• Composite Function: A function composed of two or more functions, such as $f(g(x))$.
• Chain Rule: A fundamental rule in calculus used to differentiate composite functions.
• Adjust and Compensate: A technique used to handle coefficients that do not match exactly in integration.
• Main Function: The primary function being integrated, often identified by its structure.
• Inverse Operations: Operations that undo each other, such as differentiation and integration.

#Summary and Key Takeaways

#Summary

• Integrating composite functions involves using the reverse chain rule.
• Identify the main function and adjust and compensate for coefficients.
• Special cases include functions raised to a power and integrals of the form $f'(x)/f(x)$.
• Always check your work by differentiating if possible.

#Key Takeaways

• Recognize patterns in integrals to simplify the process.
• Use the adjust and compensate method for coefficients.
• Practice spotting the main function and applying the reverse chain rule.

#Exam Strategy

1. Identify Patterns: Quickly recognize patterns like $f'(x)/f(x)$ to speed up integration.
2. Check Work: If time allows, differentiate your result to ensure it matches the original function.
3. Adjust and Compensate: Don't forget to adjust and compensate for coefficients.

#Real-World Applications

Integrating composite functions is useful in physics for calculating areas, volumes, and solving differential equations that model real-world phenomena.

Encouragement: Keep practicing! The more you work with these integrals, the more intuitive they will become. Good luck!