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.

∫ x⁴e^x/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.

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,

∫ x⁴e^x/2dx = (x⁴)(2e^x/2) - (4x³)(4e^x/2)+(12x²)(8e^x/2)-(24x)(16^x/2)+(24)(32e^x/2)

= 2x⁴e^x/2 - 16x³e^x/2+96x²e^x/2-384xe^x/2+768e^x/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.