Illustration of integration by parts - integration technique, Mathematics

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Example of Integration by Parts - Integration techniques

Some problems could need us to do integration by parts many times and there is a short hand technique that will permit us to do multiple applications of integration by parts quickly and easily.

Illustration: Evaluate the following integral.

∫ x4ex/2 dx

Solution

We start off by selecting u and dv as we always would. Though, in place of calculating du and v we put these into the following table. After that we differentiate down the column corresponding to u till we hit zero.  In the column that is corresponding to dv we integrate one time for each entry in the first column.  There is as well a third column that we will describe in a bit and it all time starts with a "+" and after that alternates signs as displayed below.

2380_Illustration of Integration by Parts - Integration technique.png

Here, multiply along the diagonals that are displayed in the table. In front of each product put the sign in the third column which corresponds to the u term for this product.  In this type of case this would give,

∫ x4ex/2dx = (x4)(2ex/2) - (4x3)(4ex/2)+(12x2)(8ex/2)-(24x)(16x/2)+(24)(32ex/2)

= 2x4ex/2 - 16x3ex/2+96x2ex/2-384xex/2+768ex/2+c.

We've got the integral.  This is much easier than writing down all the various u's and dv's that we'd have to do otherwise.


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