Partial Fractions: Integration by Rule of Everywhere, Not There!

When it comes to solving integration problems involving partial fractions, students often feel the process is long and tedious. But what if there’s a shortcut that’s not just fast, but also kind of fun?

Presenting the Rule of “Everywhere, Not There!” — a trick that can help you make partial fractions in less than 30 seconds!

This method of integration especially works when the given rational function is a proper fraction, and the denominator factors into distinct linear factors. Let’s break it down in a simple way.

What is Partial Fraction Decomposition?

When we are asked to integrate a rational function like:

We split it into simpler parts:

The goal is to find values of A and B. Normally, we’d multiply both sides by the denominator and equate different coefficients like the following:

And so, we can get the values of A and B, but here is a faster and a smarter way to do it…

Introducing the Rule of “Everywhere, Not There!”

This shortcut or you can say a trick helps you find A, B, C (etc.) quickly for rational expressions with partial fractions. Here’s how it works:

Consider the linear factors in the denominator one-by-one, you can put them equal to zero orally, as explained below:

For example, we want to decompose the rational expression:

  

We assume:



Now, instead of solving a system of equations (i.e. by equationg the coefficients as we normally do), apply the Rule of Everywhere, Not There:

Step 1: Put (i. e. root of ) everywhere in the given rational expression , except in the factor itself in denominator! So, one of the partial fractions will be:

  (means )

So,



Step 2: Now put (i. e. root of ) everywhere in the given rational expression , except in the factor itself in denominator!

  (means )

So that,

and we get:

Now, integration becomes easy:

For more understanding you can even watch a complete video on YouTube here.

When Does This Work?

The “Rule of Everywhere, Not There!” works beautifully and quickly when:

• The denominator has distinct linear factors like (x – a)(x – b)(x – c), etc.
• The numerator degree is less than the denominator (i.e., a proper rational function)

But this trick doesn’t work directly for:

• Repeated roots like (x – a)²
• Or non-linear factors (e.g., quadratics that can’t be split like x² + x + 1 or x² + 1)

Because those cases involve additional terms which need systems or equating coefficients.

Example: Rule of Everywhere, Not There! – For 3 Linear Factors

Decompose:

  

Assume:

Step 1: Put (i. e. root of ) everywhere in the given rational expression , except in the factor itself in denominator. So the first partial fraction will be:

  (means )

Step 2: Put (i. e. root of ) everywhere in the given rational expression  , except in the factor itself in denominator. So the corresponding partial fraction will be:

  (means )

Step 3: Put (i. e. root of ) everywhere in the given rational expression , except in the factor itself in denominator. So the 3rd partial fraction will be:

  (means )

Final Decomposition:

  which can be easily integrated now.

3 Benefits of Making Partial Fractions Using the Shortcut

1. Saves Precious Time in MCQs
When solving multiple-choice questions, especially in competitive exams, you don’t need to show steps. This trick lets you quickly plug in smart values (like roots of the denominator) and directly find constants like A, B, and C—cutting down time drastically and boosting speed.


2. Verifies Results in Step-by-Step Questions
In exams where you’re required to show the full process (like equating coefficients in 2- or 3-mark questions), this shortcut acts as a great double-check tool. You can confirm the values you’ve found through detailed steps by plugging in special x-values and ensuring consistency.


3. Keeps Your Focus on Integration
While practicing integration problems, this trick helps you breeze through the partial fraction stage. Instead of spending time solving systems of equations to find constants, you can use this shortcut and get to the actual integration faster—which is often the more critical part of the problem.

Final Thoughts

The Rule of “Everywhere, Not There!” brings back nostalgia for many who discovered it in Class 12 — that feeling of fascination when a long process becomes easy and fast. So, the next time you face an integration problem using partial fractions, remember this little trick, and save big time!

Try it out on other expressions, and you’ll see why it’s a favorite among math students. It’s fast, simple, and surprisingly fun!