example of integration by parts involving algebraic manipulation

For some integrals which require integration by parts, it may be to treat the integral like a variable and solve for it. For example, consider the integral


Using integration by parts with the substitutions


we obtain


Using integration by parts on the integral on the right hand side with the substitutions


we obtain


The “trick” is to add excosxdx to both sides of the equation. Some people find this concept surprising at first sight, especially since most people who are taking calculus for the first time do not use equations when showing their work for integration. For integrals such as excosxdx, writing out an equation is essential.

After adding excosxdx to both sides of the above equation, we will need a +C on the right hand side. Thus, we obtain


Therefore, we can figure out what excosxdx is by dividing both sides by 2, which yields


On the other hand, since C is an arbitrary constant, we generally write


with the understanding that the constant C in the final equation may not have the same value as C appearing in equations in previous steps.

Title example of integration by parts involving algebraic manipulation
Canonical name ExampleOfIntegrationByPartsInvolvingAlgebraicManipulation
Date of creation 2013-03-22 17:39:52
Last modified on 2013-03-22 17:39:52
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 10
Author Wkbj79 (1863)
Entry type Example
Classification msc 97D70
Classification msc 26A36
Related topic ALectureOnIntegrationByParts