Simple Integration Techniques You Can Learn Fast

🔥 Simple Integration Techniques You Can Learn Fast

Right—so, integration. It’s one of those topics where you can almost see the panic ripple across the room when a weird-looking integral appears on the board. And honestly, I get it. Some integrals look… unpleasant. But a lot of the time, you don’t need anything fancy; you just need to see what the examiner is hinting at. Hang on—before we dive into the messy stuff, let me just say this: most of these questions follow a pattern, and if you’re gradually building your A Level Maths techniques, it’s a pattern you’ll start spotting quicker than you think.

And hey, I’ve watched so many Year 12s go from “no idea what’s happening here” to “oh, that’s basically chain rule backwards.” So don’t stress. Let’s talk like a human, not like a textbook.

🔙 Previous topic:

If you missed the step before this, we just looked at How to Use the Second Derivative to Identify Max/Min Points — worth a read first, because integration makes more sense when differentiation feels solid.

📘 Exam Context

Examiners absolutely love substitution. I swear it’s like their favourite game: make an integral look scarier than it really is. But they’re also sneaky—sometimes the correct method is reverse chain rule, but they sprinkle misleading bits that make you think you need a full substitution.
A lot of scripts lose marks because people:

pick the wrong substitution
pick a substitution too early
forget to rewrite the whole integral
or just… panic and start simplifying everything until the structure disappears

The trick is slowing down enough to see the shape of the thing.

📐 Problem Setup

Let me start you with one that’s practically waving at you:
$\int (3x^2 + 1)^5 (6x), dx$
Don’t do anything yet—just look at it. This is what I call a “please use the reverse chain rule” question.
We’ll unpack it properly in a minute.

🔸 Reverse Chain Rule — the easy wins students forget to take

Okay—reverse chain rule is simply chain rule… backwards. But don’t think of it formally; think of it like this: if you see a bracket and somewhere nearby its derivative, the integral is basically done already.
The pattern is:

$\int f(g(x)), g'(x), dx = F(g(x)) + C$

Take our example:
$\int (3x^2 + 1)^5 (6x), dx$
Inside: $3x^2 + 1$

Outside: some power
Derivative of inside:

$6x$ — right there, practically waving at you.

So the integral becomes:
$\frac{(3x^2 + 1)^6}{6} + C$

No drama. No substitution. No rearranging. Just seeing the structure.

Let me pause—because I watch students skip this easy method all the time and instead do substitution, which works but takes four times longer and introduces opportunities to slip.

🟦 When reverse chain rule nearly works… but not quite

Example:
$\int (2x + 5)^4 , dx$

Now, the derivative of $2x + 5$ is $2$ , and that’s… not there. But we can “nudge” the integral:

$\int (2x + 5)^4 , dx$
$= \frac{1}{2}\int (2x + 5)^4 (2) , dx$
$= \frac{(2x + 5)^5}{10} + C$

This is still reverse chain rule. You’re just being clever with constants. Students sometimes overthink this part. Don’t. Constants are flexible tools, not traps.

🟢 Full Substitution — when the integrand won’t cooperate

Now, substitution is for the integrals where that nice “inner function and its derivative” thing just isn’t happening.

Try this:
$\int x\sqrt{x^2 + 4} , dx$

Inside: $x^2 + 4$
Derivative: $2x$ (close-ish to the $x$ we have)

Let:
$u = x^2 + 4$
$\frac{du}{dx} = 2x \Rightarrow du = 2x , dx$

So:
$x , dx = \frac12 du$

Rewrite the whole integral:
$\frac12 \int u^{1/2} , du$

Integrate:
$\frac12 \cdot \frac{u^{3/2}}{3} + C$
$= \frac{(x^2 + 4)^{3/2}}{3} + C$

Clean. Predictable. And absolutely substitution.

Actually, a quick anecdote—one of my old students used to try three different substitutions before writing anything down. Don’t do that. If the structure gets simpler after substituting, it’s the right one. If it gets uglier… undo it.

🟣 Choosing the correct substitution (the part teachers always wish students would slow down for)

Here’s my “don’t overthink it” checklist:

Pick the obvious inside
If there’s a bracket, start there.
If there’s a fraction, try substituting the denominator.
If there’s a √ something, use the thing inside the root.
If the substitution makes the algebra worse, abandon it early.

Let me just say this plainly: the best substitution almost always makes the integral shorter, not longer. If your u-expression grows three extra branches, something went wrong.

💬 Mid-topic check

Just asking these questions makes life easier:

“Is the derivative of the bracket nearby?”
“Will this substitution simplify the expression?”
“Am I forcing a method because I’m panicking?”
“Could this be a reverse chain rule pretending to be substitution?”

Being method-agnostic is part of your A Level Maths revision essentials—don’t tie yourself to one technique.

⭐ Worked Example — trig flavour (students find these reassuring)

$\int \cos(3x^2), 6x, dx$
Inner: $3x^2$
Derivative: $6x$
Everything’s lined up.
$\sin(3x^2) + C$
This is why recognising the reverse chain rule saves so much time.

🔺 Worked Example — when substitution is non-negotiable

(You can’t cheat this one.)

$\int \frac{x}{x^2 + 1}, dx$

Inner: $x^2 + 1$
Derivative: $2x$

We’ve got something close but not perfect.

Let:
$u = x^2 + 1$
$du = 2x, dx$
$x, dx = \frac12 du$

(You can’t cheat this one.)

$\int \frac{x}{x^2 + 1}, dx$

Inner: $x^2 + 1$
Derivative: $2x$

We’ve got something close but not perfect.

Let:
$u = x^2 + 1$
$du = 2x, dx$
$x, dx = \frac12 du$

Now:

$\int \frac{x}{x^2 + 1} , dx$
$= \frac{1}{2} \int \frac{1}{u} , du$
$= \frac{1}{2} \ln|u| + C$
$= \frac{1}{2} \ln(x^2 + 1) + C$

That’s the kind of question examiners give when they want to see if you genuinely understand substitution.

⚠ Common Errors & Exam Traps

Substituting something too “big” (like the whole numerator + denominator)
Forgetting to rewrite dx properly
Leaving bits of x in after converting to u
Back-substituting incorrectly
Using substitution on an integral screaming “reverse chain rule!”
Simplifying away the structure before choosing a method
Picking a u that makes the integral explode into something awful

Actually, one more thing: some students treat substitution like a ritual—“pick anything and try it.” No. Substitution is only good when it simplifies the structure.

🌍 Real-World Link

You might not believe me now, but substitution appears everywhere:

population modelling
exponential decay problems
changing variables in physics (classic)
probability distributions
current/voltage models in circuits
volumes of solids
even in data smoothing methods

Whenever one variable depends on another in a tangled way, substitution untangles it.

🚀 Next Steps

If you want substitution and reverse chain rule to feel like second nature rather than guesswork, the A Level Maths Revision Course for fast improvement walks through loads of examples with the kind of “here’s what students usually mess up” commentary teachers never write in textbooks.

📏 Recap Table

Reverse chain rule → inner + its derivative present
Substitution → used when the derivative isn’t available cleanly
Good substitution → makes the integral shorter
Bad substitution → complicates everything
Always convert dx and rewrite the whole integral
Don’t forget to back-substitute

👤Author Bio – S Mahandru

I’ve taught A Level Maths long enough to see every creative wrong substitution imaginable. Once you learn to spot the structures, integration becomes surprisingly forgiving—almost fun, even. (Almost.)

🧭 Next step:

If substitution and reverse chain rule are feeling steadier now, the next step is Partial Fractions for Integration – Full A Level Guide, where integrals stop looking friendly and start needing structure.

❓ FAQ Section

Q1: How do I know if I should use substitution?

If the reverse chain rule doesn’t fit, or if simplifying reveals a clear “inner function,” substitution is usually your next move. A good clue is when differentiating the inside gives a piece you can find elsewhere in the integrand. When that pattern appears, substitution tends to unlock the problem quickly rather than complicate it.

Q2: Does the substitution have to be perfect?

No — it only needs to make the integral simpler than when you started. People imagine there is one correct substitution, but often several will work fine. If your choice reduces clutter and makes the structure more visible, that’s a successful substitution.

Q3: What if I choose the wrong substitution?

You’ll feel it immediately — the algebra gets worse instead of better. That’s not failure, it’s feedback, and backing out is part of normal working. Undo it, step back, and choose the more natural inner function — it’s how intuition develops.

Simple Integration Techniques You Can Learn Fast