Techniques of Integration in Calculus II

Explore various techniques of integration in Calculus II such as basic integration formulas, simplifying substitutions, completing the square, expanding powers with trigonometric identities, and eliminating square roots. Examples and solutions are provided to help understand these integration methods.

• Calculus
• Integration
• Techniques
• Trigonometric Identities
• Mathematics

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1. Calculus II Lecture #5 Techniques of Integration Civil Engineering Department College of Engineering Mustansiriayah University May 2020 1 Calculus II, Lecture #5 05/08/2024

2. Basic Integration Formulas Basic Integration Formulas 1. ?? = ? + ? 2. ? ?? = ?? + ?, (any number k) 3. (?? + ??) = ?? + ?? 4. ???? =??+1 ?+1+ ? (? 1) ?? ?= ln ? + ? 5. 6. sin??? = cos? + ? 7. cos??? = sin? + ? 8. sec2??? = tan? + ? 9. csc2??? = cot? + ? 10. sec?tan??? = sec? + ? 2 Calculus II, Lecture #5 05/08/2024

3. Basic Integration Formulas Basic Integration Formulas 1. csc?cot??? = csc? + ? 2. tan? ?? = ln cos? + ? = ln sec? + ? 3. cot? ?? = ln sin? + ? = ln csc? + ? 4. ???? = ??+ ? ?? ln ?+ ? 5. ???? = (? > 0,? 1) 3 Calculus II, Lecture #5 05/08/2024

4. Making a simplifying substitution Making a simplifying substitution Example 1: Evaluate the following integral 2? 9 ?2 9? + 1 ?? Solution: Let: ? = ?2 9? + 1 ,?? = 2? 9 2? 9 ?2 9? + 1 ?? = ?? ?= ? 1 2?? ? 1 1 2+ 1 2+1 = + ? = 2? 1 2+ ? = 2 ?2 9? + 1 + ? 4 Calculus II, Lecture #5 05/08/2024

5. Completing the Square Completing the Square Example 2: Evaluate the following integral ?? 8? ?2 Solution: 8? ?2= ?2 8? = ?2 8? + 16 + 16 = 16 ? 42 ?? ?? ?? 8? ?2= 16 ? 42= ?2 ?2 = sin 1? + ? ? = sin 1? 4 + ? 4 5 Calculus II, Lecture #5 05/08/2024

6. Expanding a Power and Using Trigonometric Identity Expanding a Power and Using Trigonometric Identity Example 3: Evaluate the following integral sec? + tan?2 ?? Solution: sec? + tan?2 ?? = (sec2? + 2sec? tan? +tan2?) ?? But, sec2? = tan2? + 1, therefore, (sec2? + 2sec?tan? +tan2?) ?? = (sec2? + 2sec? tan? +sec2? 1) ?? = 2 sec2? ?? + 2 sec? tan? ?? 1 ?? = 2tan? + 2sec? ? + ? 6 Calculus II, Lecture #5 05/08/2024

7. Eliminating a Square Root Eliminating a Square Root Example 4: Evaluate the following integral ? 4 1 + cos4? ?? 0 Solution: cos2? =1 + cos2? 1 + cos2? = 2cos2? or 2 With the same identity the angle x can be doubled:1 + cos4? = 2cos22? ? 4 ? 4 2cos22? ?? 1 + cos4? ?? = 0 0 ? 4 = 2 cos2? ?? 0 ? 4 sin2? 2 1 2 0 = 1 = 2 = 2 2 0 7 Calculus II, Lecture #5 05/08/2024

8. Separating a fraction Separating a fraction Example 5: Evaluate the following integral 3? + 2 1 ?2 ?? Solution: The second of these integrals will be solved as: 3? + 2 1 ?2 ?? = 3 ??? 1 ?2 + 2 ?? 1 ?2 ?? 1 ?2= 2sin 1? + ?2 2 ? = 1 ?2,?? = 2??? ??? = 1 2?? The combination of these results and renaming C1+ C2as C will give: The first of these integrals will be solved as: 1 2?? ? ??? 1 ?2 = 3 1 2 1 2 = 3 3? + 2 1 ?2 ?? = 3 1 ?2+ 2sin 1? + ? 2 ? 1 3 2 ?? = 3 ? + ?1= 3 1 ?2+ ?1 2 8 Calculus II, Lecture #5 05/08/2024