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Derivation of the integration by parts formula

The integration by parts theorem also referred to as “partial integration” is a theorem which relates to the integral of a product of a function to the integral of its anti-derivative and derivative. The integration by parts theorem is mostly used in transforming the anti-derivative of a product of a function to an anti-derivative in which a solution can be found easily. The integration by parts rule can be derived in just one line by integrating the differentiation product rule. In simple terms, integration by parts is the integration form of the product differentiation rule. The main idea of integration by parts is to alter an integral that you cannot solve into a simple product minus the integral that you can solve. Its formula is:. This implies that the other part of the product should be related to the derivative. And in case it is not, then substitution will not be the right approach. However, in this particular case it works, so you are on the right track.

Some of the cases where integration by parts will be useful is in the logarithmic function ln x and in the first four inverse trigonometric functions (arccos x, arctan x, arcsin x and arccotx). Above these cases, using integration by parts is very important for integrating the product of more than one type of function. For instance: x ln x, x2 sin x, x arcsec x and ex cos x. Note that in each case you can easily know the product of functions since the variable x appears several times in the function. Anytime you are faced with integrating the product of functions, you must consider variable substitution before you decide to use integration by parts. For example, x cos (x2) can be solved by substitution and not integration by parts. And when you determine that you need to use integration by parts, your next step will be to split the function and assign the variables u and dv. To assign variables to u and dv you have to always select the first function to equal to u and the remaining set of the product (including dx) to be equal to dv.

You can use integration by parts to integrate log functions, inverse trigonometric forms, a log composed with an algebraic equation, algebraic equations multiplied by sine, cosine or exponential, sine multiplied by exponential, cosine multiplied by an exponential and several others. The most basic rule in using integration by parts is determining which term to integrate and which term to differentiate.

With a lot of practice in using integration by parts, you will be able to easily determine when to use substitution and when to use integration by parts. And there are problems that can be solved by substitution and by integration by parts and there are others that cannot be solved by either substitution or integration by parts but that is covered in another course in the field of calculus.


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