Partial Fractions for Integration

🔥 Partial Fractions for Integration – Full A Level Guide

Okay — partial fractions. The first time most students see these, the reaction is somewhere between a sigh and a quiet “Why are we doing this again?” And honestly, fair. It looks like algebra for the sake of algebra. But the real reason is this:

You cannot integrate many rational expressions until they’re split first.
Examiners know this — they hide integration marks behind the algebra.

If you’re working through your A Level Maths walkthroughs, this is one of the turning-point skills: the moment you can split expressions, integrals that once looked horrifying suddenly relax.

So — let’s go slow, classroom-speed, a few pauses, no sterile textbook tone.

🔙 Previous topic:

If you’d like to loop back and see where substitution and reverse chain rule first unlocked these integrals, revisit Integration Techniques Made Easy (Reverse Chain Rule & Substitution) — the two ideas slot together perfectly once you’ve seen both sides.

📘 Exam Context

Examiners use partial fractions in exactly two ways:

Straight algebra:
“Split this into partial fractions.”
Integration without warning:
They don’t tell you to split — but you’ll never solve it unless you do.

The most common fail-point?

Students try to split before checking the denominator structure.

Different denominator types → different templates → different technique
Choose wrong once → everything breaks

We’re not rushing — this section is the whole game.

📐 Problem Setup

Warm-up example:

$\frac{5x + 3}{(x+1)(x-2)}$

Goal structure:

$\frac{A}{x+1} + \frac{B}{x-2}$

We’ll work through it properly in a moment — no algebra panic yet.

🔶 Step 1 — Identify the denominator type (THE deciding move)

There are three partial-fraction cases at A Level:

1️⃣ Distinct linear factors

e.g.

$\frac{5x+3}{(x+1)(x-2)}$ →
$\frac{A}{x+1} + \frac{B}{x-2}$

2️⃣ Repeated linear factor

e.g.

$\frac{7}{(x-4)^2}$ →
$\frac{A}{x-4} + \frac{B}{(x-4)^2}$

3️⃣ Irreducible quadratic

e.g.

$\frac{4x-1}{x^2+3}$ →
$\frac{Ax+B}{x^2+3}$

Misidentify this once and the whole question collapses.

💡 Step 2 — Distinct Linear Factors (the nicest one)

Start with our example:

$\frac{5x + 3}{(x+1)(x-2)}$

Write:

$\frac{A}{x+1} + \frac{B}{x-2}$

Multiply through:

$5x + 3 = A(x-2) + B(x+1)$

Now plug in smart values:

If $x=2$ :

$5(2) + 3 = 3B \Rightarrow B=\frac{13}{3}$

If $x=-1$ :

$5(-1) + 3 = -3A \Rightarrow A=\frac{2}{3}$

Final split:

$\frac{5x + 3}{(x+1)(x-2)} = \frac{2}{3(x+1)} + \frac{13}{3(x-2)}$

This one integrates beautifully later.

🔷 Step 3 — Repeated Linear Factors

Example:

$\frac{7}{(x-4)^2}$

Correct form:

$\frac{A}{x-4} + \frac{B}{(x-4)^2}$

Multiply:

$7 = A(x-4) + B$

Substitute $x=4$ → $B=7$
Compare coefficients → $A=0$

So in this case no change needed:

$\frac{7}{(x-4)^2} = \frac{7}{(x-4)^2}$

Repeated factors get trickier in integration later because you often get a mix of logs + power terms — but the template must still match.

🟦 Step 4 — Irreducible Quadratic Factors

Example:

$\frac{4x – 1}{x^2 + 3}$

Correct decomposition:

$\frac{Ax + B}{x^2 + 3}$

Multiply & compare:

For $x$ → $A = 4$
Constant → $B = -1$

You cannot split this into two linear pieces — it stays as one.

🧠 Integrating After Splitting (where everything suddenly feels easier)

Now integrate:

$\int \frac{5x+3}{(x+1)(x-2)} , dx$

Using our split:

$\int \left(\frac{2}{3(x+1)} + \frac{13}{3(x-2)}\right) dx$

Integrate each term:

$\frac{2}{3}\ln|x+1| + \frac{13}{3}\ln|x-2| + C$

That’s it — algebra did 90% of the work.
Integration is now just logs.

Let me pause — this is the moment students go:

“OH — partial fractions isn’t the maths.
It’s the door to the integration.”

Exactly.

And if you’re building this skill as part of your broader A Level Maths revision guidance, this is where integration confidence jumps fast — because the algebra stops feeling random and starts feeling purposeful.

🟩 Side Skill — Improper Fractions (people forget this constantly)

If the numerator’s degree is ≥ denominator, divide first.

Example:

$\frac{x^2 + 5}{x + 1}$

Polynomial division:

$x^2+5 = (x+1)(x-1) + 6$

So:

$\frac{x^2+5}{x+1} = x-1 + \frac{6}{x+1}$

Then integrate.

Forget this → meltdown.

⚠ Common Errors & Exam Traps

Using wrong template for denominator
Forgetting irreducible quadratics exist
Not dividing when improper
Leaving stray x-terms after substitution
Assuming all factors are linear
Integrating before rewriting (disaster)
Expanding way too early
Cancelling terms that aren’t common factors

If one of those stung — good. That means fixable.

🌍 Real-World Link

Partial fractions appear in:

Laplace transforms
Probability density functions
Logistics and population modelling
Circuit response curves
Mechanics of rational motion
Control systems
Any rational function in applied maths

Wherever there’s a rational expression, someone somewhere is breaking it apart.

🚀 Next Steps

If partial fractions felt like a door opening, and you’d like more integration practice where algebra + calculus lock together cleanly, the A Level Maths Revision Course that explains everything walks through repeated factors, quadratics, division, and exam-style integrations step-by-step.

📏 Recap Table

Case	Template
Distinct linear	$\frac{A}{x+a}+\frac{B}{x+b}$
Repeated linear	$\frac{A}{x+a}+\frac{B}{(x+a)^2}$
Quadratic	$\frac{Ax+B}{\text{quadratic}}$
Improper	divide first

👤Author Bio – S Mahandru

I’ve spent over a decade teaching A Level Maths and marking scripts where students collapse perfectly solvable integrals by splitting incorrectly. Once you see denominators like categories instead of chaos, partial fractions stop being a monster and start being a tool.

🧭 Next step:

If you’d like to circle back and reinforce the algebra → calculus connection, the next step is re-visiting Integration Techniques Made Easy, where reverse chain rule and substitution start to feel instinctive rather than forced.

❓ FAQ

Q1: How do I know if a quadratic is irreducible?

Check the discriminant. If it’s negative, stop trying to factor — it won’t. That’s your signal to switch to $\frac{Ax+B}{\text{quadratic}}$ and move on without wasting minutes.

Q2: What if I choose the wrong partial-fraction template?

You’ll feel it immediately — coefficients won’t solve cleanly, algebra grows teeth, constants misbehave. That’s not failure — that’s feedback. Switch to the template that matches the factor type and it usually collapses neatly.

Q3: Do all partial fractions lead to logarithms?

Linear ones nearly always do. Quadratics lean toward arctan or a log-plus-arctan mix depending on structure. Think of logs as the baseline track — quadratics add harmony.

Partial Fractions for Integration