Integrating Reverse Chain Rule and Substitution in Calculus

Explore the concepts of reverse chain rule and substitution in integration through worked examples and practice questions involving trigonometric functions. Enhance your skills with interactive narration and practical exercises. Dive into the world of calculus with a silent teacher guiding you through intelligent practice sessions.

• Calculus
• Integration
• Reverse Chain Rule
• Substitution
• Trigonometry

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1. Integration: Integration: Reverse chain rule Reverse chain rule or substitution or substitution Silent Teacher Intelligent Practice Your Turn Narration 6? ?2 43?? sin?cos??? Practice

2. Worked Example Your Turn ? ?2+ 12?? ?2?3+ 12?? ? ?2 ?2+ 1?? ?3+ 1?? @DrChris_Baker

3. 2?3 ?2 3?+??? 1. ? ?2 62?? 11. 2. sin2?cos??? ? 1.5 ?2 3?+??? 12. 3. sin5?cos??? 13. 2? 3 ??2 3?+2?? 4. ?cos ?2 3 ?? 14. 2? 3 sin ?2 3? + 2 ?? 5. ?cos ?2 3 ?? 3?2 4?+3 (?3 2?2+3? 2)?? 15. 6. ?2cos ?3 3 ?? 6?2 8?+6 (?3 2?2+3? 2)?? 16. 7. ?2cos ?3 3 ?? ?2? 3 ?2? 3 2 ?? ? 17. ?2 3?? 8. 18. tan3?sec2? ?? ?2 ?3 3?? 9. 20. ???4??????2 ? ?? 2? 3 ?2 3?+2?? 10. @DrChris_Baker 21. tan???

4. 1. sin?cos??? = 1 2cos2? + ? =1 2sin2? + ? 2. sin2?cos??? =1 3sin3? + ? 3. sin5?cos??? =1 6sin2? + ? 4. ?cos ?2 3 ?? =1 2sin(?2 3) + ? 5. ?cos ?2 3 ?? =1 2sin(?2 3) + ? 6. ?2cos ?3 3 ?? =1 3sin(?3 3) + ? 7. ?2cos ?3 3 ?? =1 3sin(?3 3) + ? 2? 3 ?2 3?+2?? = ln ?2 3? + 2 + ? ?2 3?? =1 ? 2ln|?2 3| + ? 10. 8. ?2 ?3 3?? =1 3ln|?3 3| + ? 9. @DrChris_Baker

5. 20. ???4??????2 ? ?? = 1 5cot5? + ? 2? 3 ?2 3?+??? = ln ?2 3? + 2 + ? 11. 21. tan??? = ln sec? + ? ? 1.5 ?2 3?+??? =1 2ln ?2 3? + ? + ? 12. 13. 2? 3 ??2 3?+2?? = ??2 3?+2+ ? 14. 2? 3 sin ?2 3? + 2 ?? = cos ?2 3? + 2 + ? 3?2 4?+4 (?3 2?2+3? 2)??= ln ?3 2?2+ 3? 2 + ? 15. 6?2 8?+6 (?3 2?2+3? 2)?? = 2ln ?3 2?2+ 3? 2 + ? 16. ?2? 3 ?2? 3 2 ?? =1 2ln|?2? 3 2| + ? 17. 18. tan3?sec2? ?? =1 4tan4? + ? @DrChris_Baker