Integration techniques: Integral by parts (verify the result with derivative)
The integration by parts is a very important integration technique within the integration techniques, the importance of this technique lies in the fact that it can be applied to a great variety of functions that are in the integrand, but above all they are very useful in integrands where multiplied functions are found, such as algebraic functions, transcendental functions, among others. For example, integration by parts works with integrals of the type:
Theorem 8.1. Integration by parts
If u and v are functions of x and have continuous derivatives, then:
This theorem expresses the original integral in terms of the other integral. Depending on the choice of u and dv, one thing to note is that the second integral may be easier to evaluate than the original one, since the choice of u and dv is important in integration by the process of parts, the following strategies are provided:
[1] Try to take as dv the most complicated portion of the integrand that conforms to a basic rule of integration and as u the remaining factor of the integrand.
[2] Try to take as u the portion of the integrand whose derivative is a function simpler than u, and as dv the remaining factor of the integrand.
Example of how to calculate a piecewise integral
Solve the following piecewise integral:
To solve this piecewise integral we must select from the integrand Ln(x) as the part that is easy to derive and dx as the part that is easy to integrate, as follows:
We now proceed to derive u and solve the integrals of dv and dx, as follows:
Now we apply theorem 8.1:
As follows:
As the objective of this post is to solve an integral by the piecewise method and check the result with derivative, then we take the result we got from the integration and derive it and compare if it gives us the same as the integrand:
There we were able to verify that if we derived the result of the integral, the result obtained is that of the integrand of the piecewise integral that was solved.
Bibliographic Reference
Calculus with Analytic Geometry by Ron Larson, Robert, P. Hostetler and Bruce H. Edwards. Volume I. Eighth Edition. McGraw Hill. Año 2006