MATH 212 combinepdf - Intergral Calculus Basic Integration of Formulas 1 Basic Integration Formulas This module deals with Integration of

MATH 212 combinepdf - Intergral Calculus Basic...

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Intergral Calculus Basic Integration of Formulas 1 Basic Integration Formulas This module deals with Integration of Transcendental Functions such as Trigonometric Functions, Logarithmic Functions, and Exponential Functions. It includes the introduction of the formulas of integration for each function. Course Module Objectives: At the end of this module, the learner should be able to: 1. Identify u, n, CF and du 2. Memorize the basic integration formulas for trigonometric functions 3. Apply the appropriate formula 4. Evaluate the integrals Integration of Transcendental Functions Formulas for Integration: A. Integration of Trigonometric Functions 1. ∫ cos ??? = sin ? + ? 2. ∫ sin ? ?? = − cos ? + ? 3. ∫ ??? 2 ? ?? = tan u + ? 4. ∫ ??? 2 ? ?? = − cot ? + ? 5. ∫ sec ? tan ? ?? = sec ? + ? 6. ∫ csc ? cot ? ?? = − csc u + c 7. ∫ tan ? ?? = ln sec ? + ? = − ln cos ? + ? 8. ∫ ??? ? ?? = ln sin ? + ? 9. ∫ sec ? ?? = ln(sec ? + tan ?) + ? 10. ∫ ???? ?? = ln(csc ? − cot ?) + ? B. Integrals Yielding Logarithmic Functions 11. ?? ? = ln ? + ? ; ? > 0 C. Integration of Exponential Functions 12. ∫ ? ? ?? = ? ? + ? 13. ∫ 𝑎 ? ?? = 𝑎 𝑢 ln 𝑎 + ? D. Integration Yielding Inverse Trigonometric Functions 14. ?? √𝑎 2 − ? 2 = 𝑎???𝑖? ? 𝑎 + c 15. ?? ? 2 + 𝑎 2 = 1 𝑎 𝑎???𝑎? ? 𝑎 + ? 16. ?? ?√? 2 − 𝑎 2 = 1 𝑎 𝑎????? ? 𝑎 + ?
Intergral Calculus Basic Integration of Formulas 2 Recall the Power Formula 17. ∫ ? ? ?? = ? ? + ? Imperative in the integration of Trigonometric Functions is the recall of Trigonometric Identities : sin 2? = 2 sin ? cos ? ?𝑎? 2 ? = ??? 2 ? − 1 ??? 2 ? = ??? 2 ? − 1 ?𝑖? 2 ? + ??? 2 ? = 1 ?𝑖? 2 ? = 1 2 (1 − cos 2?) ??? 2 ? = 1 2 (1 + cos 2?) A. Integration of Trigonometric Functions Illustrative Examples: Example 1: ∫ ?𝑎?5𝜃?𝜃 Solution: Let u = 5𝜃 ; du = 5d 𝜃 (A correction factor CF = 1 5 is multiplied to the integrand to correct 5d 𝜃 ) = 1 5 ∫ tan 5𝜃 (5𝜃?𝜃) = 1 5 ∫ ?𝑎?? ?? Answer: = − 1 5 ln ???5𝜃 + 𝐶 Example 2: ∫ ?𝑎? 2 𝜃 ?𝜃 = ∫(??? 2 𝜃 − 1) ?𝜃 Answer: = ?𝑎?𝜃 − 𝜃 + 𝐶 Example 3: ∫ ?𝑖?2? cos 2? ?? Recall the identity 2?𝑖? ? ??? ? = sin 2? sin 2? cos 2? = 1 2 sin 4? Let u = 4t; du = 4dt; CF = 1 4 1 2 ∫ sin 4? = − 1 8 cos 4? + ? Answer: = − 1 8 ???4? Example 4: ∫ 𝑥??? 2 3𝑥 2 ?𝑥 Let u = 3𝑥 2 ; du = 6𝑥?𝑥 = 1 6 ∫ ??? 2 3𝑥 2 ∙ 6𝑥 ?𝑥
Intergral Calculus Basic Integration of Formulas 3 1 1 = 3 1 1
Intergral Calculus Basic Integration of Formulas 4 Example 10: ??? 3 𝜃 ?𝜃 1+?𝑖?𝜃 = ∫ ??? 3 𝜃 ?𝜃 1 + ?𝑖?𝜃 1 − ?𝑖?𝜃 1 − ?𝑖?𝜃 = ∫ ??? 3 𝜃 ?𝜃 ??? 2 𝜃 − ∫ (?𝑖?𝜃 ??? 3 𝜃)?𝜃 ??? 2 𝜃 Answer: = ?𝑖?𝜃 + 1 2 ??? 2 𝜃 + 𝐶
Integral Calculus Integrals Yielding Logarithmic Functions and Integration of Exponential Functions 1 Integrals Yielding Logarithmic Functions and Integration of Exponential Functions This module covers the integrals yielding logarithmic functions and exponential functions. Course Module Objectives: At the end of this module, the learner should be able to: 1. Identify u, and du 2. Memorize the basic integration formulas for exponential functions and integrals yielding logarithmic functions 3. Apply the appropriate formula 4. Evaluate the integrals

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