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Warm-up: Evaluate the integrals. 1) 2)

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Warm-up: Evaluate the integrals. 1) 2)

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Warm-up: Evaluate the integrals. 1) 2)

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Integration by Parts Section 8.2 Objective: To integrate problems without a u-substitution

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Integration by Parts When integrating the product of two functions, we often use a u-substitution to make the problem easier to integrate. Sometimes this is not possible. We need another way to solve such problems.

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Integration by Parts As a first step, we will take the derivative of

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Integration by Parts As a first step, we will take the derivative of

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Integration by Parts As a first step, we will take the derivative of

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Integration by Parts As a first step, we will take the derivative of

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Integration by Parts As a first step, we will take the derivative of

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Integration by Parts Now lets make some substitutions to make this easier to apply.

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Integration by Parts This is the way we will look at these problems. The two functions in the original problem we are integrating are u and dv. The first thing we will do is to choose one function for u and the other function will be dv.

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Example 1 Use integration by parts to evaluate

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Example 1 Use integration by parts to evaluate

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Example 1 Use integration by parts to evaluate

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Example 1 Use integration by parts to evaluate

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Example 1 Use integration by parts to evaluate

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Guidelines The first step in integration by parts is to choose u and dv to obtain a new integral that is easier to evaluate than the original. In general, there are no hard and fast rules for doing this; it is mainly a matter of experience that comes from lots of practice.

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Guidelines There is a useful strategy that may help when choosing u and dv. When the integrand is a product of two functions from different categories in the following list, you should make u the function whose category occurs earlier in the list. Logarithmic, Inverse Trig, Algebraic, Trig, Exponential The acronym LIATE may help you remember the order.

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Guidelines If the new integral is harder that the original, you made the wrong choice. Look at what happens when we make different choices for u and dv in example 1.

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Guidelines If the new integral is harder that the original, you made the wrong choice. Look at what happens when we make different choices for u and dv in example 1.

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Guidelines Since the new integral is harder than the original, we made the wrong choice.

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Example 2 Use integration by parts to evaluate

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Example 2 Use integration by parts to evaluate

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Example 2 Use integration by parts to evaluate

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Example 2 Use integration by parts to evaluate

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Example 2 Use integration by parts to evaluate

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Example 3 (S): Use integration by parts to evaluate

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Example 3 Use integration by parts to evaluate

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Example 3 Use integration by parts to evaluate

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Example 3 Use integration by parts to evaluate

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Example 3 Use integration by parts to evaluate

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Example 4 (Repeated): Use integration by parts to evaluate

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Example 4 (Repeated): Use integration by parts to evaluate

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Example 4 (Repeated): Use integration by parts to evaluate

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Example 4 (Repeated): Use integration by parts to evaluate

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Example 4 (Repeated): Use integration by parts to evaluate

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Example 4 (Repeated): Use integration by parts to evaluate

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Example 4 (Repeated): Use integration by parts to evaluate

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Example 4 (Repeated): Use integration by parts to evaluate

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Example 5: Evaluate the following definite integral

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Example 5: Evaluate the following definite integral

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Example 5: Evaluate the following definite integral

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Example 5: Evaluate the following definite integral

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Example 5: Evaluate the following definite integral

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Example 5: Evaluate the following definite integral

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Example 5: Evaluate the following definite integral

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Example 5: Evaluate the following definite integral

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Example 5: Evaluate the following definite integral

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Example 5: Evaluate the following definite integral

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Example 5: Evaluate the following definite integral

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Example 5: Evaluate the following definite integral

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Example 5: Evaluate the following definite integral

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Homework: Page 520 # 3-9 odd, 15, 25, 29, 31, 37

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