


IntegrationIntegration by PartsThis extremely useful rule is derived from the Product Rule for differentiation. The Product Rule can be written: Integrating both sides gives: This rearranges to a more useful form: where f(x) and g(x) are two continuously differentiable functions. Suppose we have an integral of the form . We can exploit the identity above by identifying one of the factors as f '(x) and the other factor as g(x). This technique is effective when:
ExampleTo calculate Let f '(x) = cos x, so integrating gives f(x) = sin x, and g(x) = x, so differentiating gives g '(x) = 1. Then using the parts rule where C is a constant of integration. We can show the process graphically: To illustrate the care required in choosing the parts, we deliberately make a bad choice for the example above. Let f '(x) = x, integrating gives f(x) = ½x^{2}, and g(x) = cos x, differentiating gives g '(x) = sin x. But when we use the rule the new integral is harder than the old one! ExerciseThe first examples we have given are very straight forward. At times it is necessary to think outside the square in order to choose f '(x) and g(x). ExampleConsider ln x dx. We can use integration by parts but need to identify the factors. ln x = 1*ln x. We let f '(x) = 1 and g(x) = ln x then integrating gives f(x) = x and differentiating gives g '(x) = . Using the parts rule, ln x dx = x ln x  x dx = x ln x  1 dx + c = x ln x  x + c.
With more complex examples, there may be more factors to choose from. Some combinations will be much more difficult than others. ExampleConsider This time we have more ways to split into 2 factors. Since the derivative of is , we can integrate whereas we can't integrate . So let f '(x) = 2 which integrates to f(x) = , and g(x) = ½x^{2}, which differentiates to g '(x) = x. Using the parts rule: Multiple Integration by Parts  ReductionSometimes it is necessary to integrate by parts more than once. By repeatedly using integration by parts, integrals such as or can be computed in the same manner, each application of the rule lowers the power of x by one. ExampleLet f '(x) = e^{x}, integration yields f(x) = e^{x}, and g(x) = x^{3}, differentiation gives g '(x) = 3x^{2}. Then using the parts rule Again let f '(x) = e^{x}, so that f(x) = e^{x}, and g(x) = 3x^{2}, which differentiates to g '(x) = 6x. Then using the parts rule again: Once again let f '(x) = e^{x}, so that f(x) = e^{x}, and g(x) = 6x, which differentiates to g '(x) = 6. Using the parts rule yet again:
Putting it all together: = x^{3}e^{x} − 3x^{2}e^{x} + 6xe^{x} − 6e^{x} + c. Periodic SolutionsAn interesting example that is often seen is: where the full integration does not need to be calculated. We use integration by parts twice. Let f '(x) = e^{x}, so that f(x) = e^{x}, and g(x) = sin x, which differentiates to g '(x) = cos x. Using the parts rule:
This new integration looks very similar to the original one, but persist. To evaluate the new integral, we use integration by parts again. Let f '(x) = e^{x}, so that f(x) = e^{x}, and g(x) = cos x, which differentiates to g '(x) = sin x. Using the parts rule:
Combining these two, results in
The same integral appears on both sides of this equation. So we can simply solve the equation for the integral: It remains only to add a constant of integration. ExerciseThere are 10 types of problems found in these exercises. Do plenty of exercises until you feel confident about all types. Inverse Trigonometric FunctionsExamplearctan(x) dx, where arctan(x) is the inverse tangent function. Let f '(x) = 1 so integrating, f (x) = x and g(x) = arctan(x). Remembering that = arctan(x) + c, g '(x) = Using the parts rule, arctan(x) dx = x arctan(x) − = x arctan(x) − ½ln(x^{2} + 1) + c. Note that we have used the method of substitution to evaluate the final integral. ExerciseCalculate arcsin(x) dx. Show answer For completeness, arccos(x) dx = xarccos(x) − + c. << Integration by Trigonometric Substitution  Integration Index 