K(ax + b) + m(cx + d) = px + q
Tabular Integration and the LIPET Scheme
Sometimes one has to perform integration by parts many times in order to arrive at an answer. Tabular integration, described below, is a mechanism designed to streamline this process.
To integrate [pic], or more formally [pic] using integration by parts, we would have [pic]. The integration by parts formula then says
[pic]
or in u, v shorthand
[pic].
We “tabularize” this process by drawing a table:
|[pic] |[pic] |
|[pic] |[pic] |
The top row of this table represents the original integrand broken into two pieces. We take a derivative as we go down the first column while we take an antiderivative as we go down the second column.
Next, we draw two kinds of arrows: A downward slanted arrow represents an ordinary product while a horizontal arrow represents an integral of a product. Furthermore, a solid arrow is positive and a dashed arrow is negative. Our rule for integration by parts now looks like
|[pic] |[pic] |
|[pic] |[pic] |
As an example, let’s use tabular integration to evaluate [pic]:
|[pic] |[pic] |
|[pic] |[pic] |
From the table, we have
[pic]
The table is a quick way to read off the first line of the above computation. Again, recall that the slanted arrow represents an ordinary product. This arrow is solid, so this product is added. The horizontal arrow represents the integral of a product; this arrow is dashed, so this integral is subtracted.
The strength of this scheme lies in the fact that this process can be extended by adding rows to the table. Again, the first row contains the factors of the original integrand. The entries in the first column are successive derivatives of the top entry in the first column. The entries in the second column are successive antiderivatives of the top entry in the second column. One simply draws slanted arrows down the table, finishing with a horizontal arrow through the last row. The arrows alternate, beginning with solid (positive.)
|[pic] |[pic] |
|[pic] |[pic] |
|[pic] |[pic] |
|[pic] |[pic] |
|[pic] |[pic] |
We use the notation [pic] to represent an antiderivative of g(x), [pic] to represent an antiderivative of [pic] and so on. The above table represents the formula
[pic]
To see this in action, let’s evaluate [pic]. Notice that antiderivatives of [pic] are very easy to compute, so it’s natural to put [pic] as the first entry in the second column, leaving [pic] as the first entry in the first column. We obtain the table
|[pic] |[pic] |
|[pic] |[pic] |
|[pic] |[pic] |
|[pic] |[pic] |
|[pic] |[pic] |
|[pic] |[pic] |
From this, we quickly obtain
[pic]
That last integral, [pic], is a fancy way to write the constant of integration, since an antiderivative of 0 is a constant.
“Integration by partial magic” sometimes occurs when you mix exponentials and trig functions in the same integral. If you were to try tabular integration, you’d never stop. On the other hand, you might end up with a copy of the same integral, which can be useful. Let’s try [pic]. Using tabular integration, we have
|[pic] |[pic] |
|[pic] |[pic] |
|[pic] |[pic] |
Our arrows would give us
[pic]
This looks decidedly unhelpful, as we have the same integral on the right-hand side as we do on the left-hand side, but notice that it is subtracted. Therefore, let’s add our integral to both sides:
[pic]
The LIPET scheme.
There is a scheme called ‘LIPET’ that’s helpful in deciding “what is u and what is dv.” Given an integrand that is the product of two pieces, place the pieces on a chart based on the following scheme:
|(L)ogarithms | |
|(I)nverse Trig | |
|(P)olynomials or (P)owers of x | |
|(E)xponentials | |
|(T)rig | |
Whichever piece is higher is assigned as u, the rest of the integrand is assigned as dv.
Try this with [pic]:
|(L)ogarithms | |
|(I)nverse Trig | |
|(P)olynomials or (P)owers of x |(2x+1) |
|(E)xponentials |ex |
|(T)rig | |
So, we’d have u = 2x + 1, dv = ex dx; du = 2 dx, v = ex:
[pic]
Let’s find [pic]. It looks like we have only one piece of the integral to put into the LIPET scheme, namely the ln x, but in this case, we can invent another piece, namely the invisible ‘1’ (this is treated as a polynomial of x for LIPET purposes.)
|(L)ogarithms |ln x |
|(I)nverse Trig | |
|(P)olynomials or (P)owers of x |1 |
|(E)xponentials | |
|(T)rig | |
Into our chart these go:
|[pic] |[pic] |
|[pic] |x |
We then have
[pic]
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