ITLR - Integration by Parts Lesson

Integration by Parts

Integration by parts is a technique for finding integrals involving a product of two functions. It is particularly effective when integrands involve a product of algebraic and transcendental functions. Just as the Substitution Rule for integration corresponds to the Chain Rule for differentiation, the integration by parts rule corresponds to the Product Rule for differentiation. The presentation below develops the rule for integration by parts.

As shown in the video, the integration by parts formula expresses one integral in terms of a second integral equation image indicator $LaTeX: \int f\left(x\right)g'\left(x\right)dx=f\left(x\right)g\left(x\right)-\int g\left(x\right)f'\left(x\right)dx$ $\int f\left(x\right)g'\left(x\right)dx=f\left(x\right)g\left(x\right)-\int g\left(x\right)f'\left(x\right)dx$ . A simpler version of this formula is obtained by letting u = f(x) and v = g(x), and correspondingly du = f '(x) dx and dv = g'(x) dx. Thus, an alternate formula for integration by parts is equation image indicator $LaTeX: \int udv=uv-\int vdu$ $\int udv=uv-\int vdu$ . The next presentation uses this simpler version of the integration by parts rule.

Criteria and Guidelines for Integration by Parts

Certain types of functions or combinations of functions are excellent candidates for the technique of integration by parts.

Integrals where functions such as natural logarithms or inverse trigonometric functions are the sole function
Integrals involving a combination of two unrelated functions (one is not the derivative of the other)

Choosing a function to represent u and a function to represent v is critical to the success of finding an integral that is simpler and easier to evaluate than the original integral. The main objective then becomes finding a function that becomes simpler when differentiated, or at least no more complex, whenever dv = g'(x)dx can be easily integrated to give v.

Although the choice of u and dv sometimes does not matter, the mnemonic LIATE has gained popularity for choosing u based on the following order:

1. L (logarithmic)

2. I (inverse trigonometric)

3. A (algebraic)

4. T (trigonometric)

5. E (exponential)

Observe how the choice of u and dv influences the application of the integration by parts rule in the next two presentations.

Repeated Integration by Parts

For certain integrals it is sometimes necessary to perform integration by parts repeatedly. View the presentation below illustrating the necessity of using repeated integration by parts.

The next presentation includes a problem requiring repeated integration as well as other examples that require only one application of integration by parts.

Tabular Method

When repeated integration by parts is necessary, it is often difficult to organize the results of each phase of the process. The tabular method illustrated in the presentation below provides an organizational tool that is particularly helpful when the n^th derivative of one of the functions is zero and for integrals of the form equation image indicator $LaTeX: \int x^ne^{ax}dx, \int x^n\sin axdx,\:\&\int x^n\cos axdx$ $\int x^ne^{ax}dx, \int x^n\sin axdx,\:\&\int x^n\cos axdx$ .

Although integration by parts is an extremely powerful and useful technique for evaluating integrals, it is important to note that integration by parts does not always work. When it does not, other techniques should be employed.

Integration by Parts Practice

Evaluate the following integrals.

Integration by Parts: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of integration by parts.

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