📚 Topic Summary
Integration by substitution, often called u-substitution, is a powerful technique to simplify integrals. The core idea is to reverse the chain rule. By carefully choosing a 'u' that represents a part of the integrand and then finding 'du', we can transform a complex integral into a more manageable one. The goal is to express the entire integral in terms of 'u' and 'du' and then solve the integral with respect to 'u', remembering to substitute back to the original variable at the end.
The key is recognizing a composite function within the integral and identifying its inner function as a suitable choice for 'u'. Practice is essential to master this skill. This worksheet provides a range of problems to help you sharpen your integration skills!
🧠 Part A: Vocabulary
Match each term with its correct definition:
| Term |
Definition |
| 1. Integrand |
A. The function resulting from integration |
| 2. Substitution |
B. The function being integrated |
| 3. Antiderivative |
C. A technique used to simplify integrals by changing variables |
| 4. Chain Rule |
D. A rule that finds the derivative of composite functions |
| 5. $du$ |
E. Derivative of the chosen 'u' variable |
(Match the terms with their definitions.)
✍️ Part B: Fill in the Blanks
Complete the following paragraph using the words provided: derivative, integral, u-substitution, chain rule, variable.
________ is a method used to simplify complex ________. It's essentially the reverse of the ________. By choosing a suitable ________, 'u', and finding its ________, 'du', we transform the original expression into a simpler ________.
🤔 Part C: Critical Thinking
Explain in your own words why choosing the 'correct' $u$ is crucial in integration by substitution. What happens if you choose a 'u' that doesn't simplify the integral?