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Section 4.5 Integration by Substitution IB Calculus Objective: Mrs. Francis In this lesson you learn how to evaluate different types of definite and indefinite integrals using a variety of methods. Date: I. Pattern Recognition (Pages 297ο299) The role of substitution in integration is comparable to the role of in differentiation. π [π(π(π))] π π = π′ (π(π))π′(π) From the definition of an antiderivative, it follows that: ∫ π′ (π(π))π′ (π)π π = π(π(π)) + πͺ Antidifferentiation of a Composite Function ∫ π′ (π(π))π′ (π)π π = π(π(π)) + πͺ Letting π = π(π) gives π π = π′ (π)π π and From the definition of an antiderivative, it follows that: ∫ π(π)π π = π(π) + πͺ π Example 1: ∫(ππ + π) (ππ)π π Example 2: ∫ π πππ ππ π π What you should learn How to use pattern recognition to find an indefinite integral Many integrands contain the essential part (the variable part) of π′(π₯) but are missing a constant multiple. In such cases, you can . ∫ π π(π)π π = π ∫ π(π)π π Example 3: ∫ π(ππ + π)π π π II. Change of Variables (Pages 300ο301) What you should learn You should learn how to integrate using substitution With a formal change of variables, you completely . ο ο²ο f (g(x))gο’ (x) dx ο½ο ο² . Example 4: a. Find ∫ √2π₯ − 1 ππ₯ b. Find ∫ π ππ2 3π₯ cos 3π₯ ππ₯ Guidelines For Making Change of Variables 1. Choose a substitution π’ = π(π₯). Usually it is best to choose the inner part of a composite function, such as a quantity raised to a power. 2. Compute ππ’ = π′ (π₯)ππ₯ 3. Rewrite the integral in terms of the variableπ’. 4. Find the resulting integral in terms of π’. 5. Replace π’ by π(π₯) to obtain an antiderivative in terms of π₯. 6. Check your answer by differentiating. III. The General Power Rule for Integration One of the most common u substitutions involves quantities in the integrand that are raised to a power. Because of the importance of the type of substitution, it is given a special name – the General Power Rule for Integration. Example 5 – Substitution and the General Power Rule a. ∫ 3(3π₯ − 1)4 ππ₯ b. ∫(2π₯ + 1)(π₯ 2 + π₯)ππ₯ c. ∫ 3π₯ 2 √π₯ 3 − 2 ππ₯ d. ∫ (1−2π₯ 2)2 ππ₯ e. ∫ πππ 2 π₯ sin π₯ ππ₯ −4π₯ IV. Change of Variables for Definite Integrals (Pages 303ο304) When using u-substitution with a definite integral, it is often convenient to rather than to convert the antiderivative back to the variable x and evaluate the original limits. Change of Variables for Definite Integrals If the function u ο½ο g(x) has a continuous derivative on the closed interval [a, b] and f is continuous on the range of g, then Example 6 – Change of Variables 1 Evaluate ∫0 π₯(π₯ 2 + 1)3 ππ₯ What you should learn How to use a change of variables to evaluate a definite integral V. Integration of Even and Odd Functions (Page 305) Occasionally, you can simplify the evaluation of a definite integral over an interval that is symmetric about the y-axis or about the origin by . Let f be integrable on the closed interval [οο a, a]. π π If f is an function, then ∫−π π(π₯)ππ₯ = 2 ∫0 π(π₯)ππ₯ If f is an function, then ∫−π π(π₯)ππ₯ = 0 π π 2 π − 2 Example 7 – Evaluate ∫ (π ππ3 π₯ πππ π₯ + π πππ₯ πππ π₯)ππ₯ What you should learn How to evaluate a definite integral involving an even or odd function