Tabular Integration: Faster & accurate way to solve repeated integration by parts

Objective of this article: To explain a technique to super-boost one’s speed and accuracy for repeated integration by parts and to present worked examples.

Assumed knowledge: High school calculus (Integration)

In calculus, we have learnt a wide variety of methods to solve integrands — using the table of integrals and it’s properties, integration by substitution (change of variable), integration by trigonometric substitution, by partial rational fractions, of rational fractions, of irrational functions, etc.

One of these many methods is integration by parts. In this, we solve the integrand using the following formula.

I will assume that we are aware how to derive this formula (if not, start with y = uv, differentiate both sides with respect to x, rearrange the rule and then integrate both sides of the equation and it should give (1)). Now, let’s say we want to solve,

First, we’ll solve this traditionally, using (1).

The formula was repeatedly used 4 times to find the final solution. For easier problems like this, it’s okay to use the integration by parts formula. But, if for example, we had x¹² instead of x⁴, the formula then has to be applied 12 times repeatedly instead of 4 to get the answer. That would make solving it very long, tiring and repetitive.

It’s like asking you to add 6 + 7 + 8 + 6 + 8 + 6 + 8 + 7 + 7 + 6 + 8 + 6 + 6 + 6 + 7 + 7 + 8 + 6 + 6 + 8 + 6 + 8 + 7 + 6 + 8 + 6 + 7 + 6 + 6 + 8 + 6 + 6 + 8 + 7 + 7 + 5 one number at a time. Isn’t it easier to solve (5 x 1) + (6 x 16) + (7 x 9) + (8 x 10)?

In the way multiplication simplifies long, repetitive additions, what if I tell you there’s a faster way to solve more longer, repetitive and complicated integration by parts problems with the bonus of higher accuracy?

Let’s take the integration by parts formula (1) and substitute u = f(x) and dv = g(x) dx. The formula can now be written as,

For simplicity, I’m removing ‘dx’,

Do you see a pattern forming here?

Take a look at each term underlined with red in the equation above. We’ll take these individual terms and create the following table:

(Let’ s denote items of derivatives column by ‘Dn’ and those of integrals column by ‘In’ where ‘n’ is the row number)

Thus, the solution of integrating f(x)g(x) would be just multiplying D1 by I2, D2 by I3, D3 by I4 and so on, as shown by the red arrows in the table above, then summing these terms and alternating signs between them. (i.e., D1.I2 - D2.I3 + D3.I4 - …)

This method is called ‘Tabular Integration’.

Let’s solve the above example again. But this time, we’ll use Tabular Integration.

Following the arrows in the above table,

Here, we’ve got the same answer in merely two steps! Much faster and with much more accuracy.

Let’s try another example so that this gets much clearer.

Now putting this in the table,

Following the arrows in the above table,

Thus, we have found a method that is so much more speedy and error-free and makes solving integrands using integration by parts so much easier! You can now open any book, worksheet or chapter with integrands to be solved using integration by parts and use the Tabular Integration method. You’ll see how simple solving them has become.