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

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.

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

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

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

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

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

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

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.

Example 1 Use integration by parts to evaluate

Example 1 Use integration by parts to evaluate

Example 1 Use integration by parts to evaluate

Example 1 Use integration by parts to evaluate

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.

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.

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

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.

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

Example 2 Use integration by parts to evaluate

Example 2 Use integration by parts to evaluate

Example 2 Use integration by parts to evaluate

Example 2 Use integration by parts to evaluate

Example 3 Use integration by parts to evaluate

Example 3 Use integration by parts to evaluate

Example 3 Use integration by parts to evaluate

Example 3 Use integration by parts to evaluate

Example 4(repeated) Use integration by parts to evaluate

Example 4(repeated) Use integration by parts to evaluate

Example 4(repeated) Use integration by parts to evaluate

Example 4(repeated) Use integration by parts to evaluate

Example 4(repeated) Use integration by parts to evaluate

Example 4(repeated) Use integration by parts to evaluate

Example 5 Use integration by parts to evaluate

Example 5 Use integration by parts to evaluate

Example 5 Use integration by parts to evaluate

Example 5 Use integration by parts to evaluate

Example 5 Use integration by parts to evaluate

Example 5 Use integration by parts to evaluate

Example 5 Use integration by parts to evaluate

Example 5 Use integration by parts to evaluate

Example 5 Use integration by parts to evaluate

Example 5 Use integration by parts to evaluate

Example 5 Use integration by parts to evaluate

Example 5 Use integration by parts to evaluate

Example 5 Use integration by parts to evaluate

Tabular Integration Integrals of the form where p(x) is a polynomial can sometimes be evaluated using a method called Tabular Integration.

Tabular Integration Integrals of the form where p(x) is a polynomial can sometimes be evaluated using a method called Tabular Integration. 1.Differentiate p(x) repeatedly until you obtain 0, and list the results in the first column.

Tabular Integration Integrals of the form where p(x) is a polynomial can sometimes be evaluated using a method called Tabular Integration. 1.Differentiate p(x) repeatedly until you obtain 0, and list the results in the first column. 2.Integrate f(x) repeatedly until you have the same number of terms as in the first column. List these in the second column.

Tabular Integration Integrals of the form where p(x) is a polynomial can sometimes be evaluated using a method called Tabular Integration. 1.Differentiate p(x) repeatedly until you obtain 0, and list the results in the first column. 2.Integrate f(x) repeatedly until you have the same number of terms as in the first column. List these in the second column. 3.Draw diagonal arrows from term n in column 1 to term n+1 in column two with alternating signs starting with +. This is your answer.

Example 6 Use tabular integration to find

Example 6 Use tabular integration to find Column 1

Example 6 Use tabular integration to find Column 1 Column 2

Example 6 Use tabular integration to find Column 1 Column 2

Example 6 Use tabular integration to find Column 1 Column 2

Example 7 Evaluate the following definite integral

Example 7 Evaluate the following definite integral

Example 7 Evaluate the following definite integral

Example 7 Evaluate the following definite integral

Example 7 Evaluate the following definite integral

Example 7 Evaluate the following definite integral

Example 7 Evaluate the following definite integral

Example 7 Evaluate the following definite integral

Homework Section 7.2 Page multiples of 3