Integration Made Simple: 5 Real A Level Questions Explained Step-by-Step

💬 “Sir, how do you know which method to use?”
That’s probably the most common question I hear in Pure lessons, right after “Do we have to show all the working?”

Integration looks innocent — neat, smooth, even friendly — until you open an A Level paper and find five different types of questions all pretending to be the same thing.

So in this post, we’re going to walk through five real A Level integration examples and talk through how to spot the method fast — because the hardest part isn’t doing the maths, it’s recognising which rule applies.

No wall of formulas here. Just logic, patterns, and a few classic “teacher moments” you’ll recognise from any good maths lesson.

🔙 Previous topic:

“Revisit differentiation patterns before simplifying integration problems.”

🧩 Why Integration Trips Students Up

Integration questions are sneaky because they blend techniques — substitution hides inside chain rule, by-parts show up when you least expect it, and exam boards love mixing trig and exponentials in one go.

I once had an Edexcel student who could integrate anything… until it appeared in context. A velocity question threw her completely. Why? She didn’t notice that displacement = ∫ velocity. Same maths, different disguise.

So let’s start spotting the disguises.

🧮 Example 1 — The Classic Power Rule (AQA’s Favourite Warm-Up)

When you see something like:

Integrate 3x² + 2x − 5

That’s your instant cue: Power rule territory.
Add one to the power, divide by the new power. Simple.

Result:

∫(3x² + 2x − 5) dx = x³ + x² − 5x + C

💬 Teacher tip:
AQA often gives one of these early to calm you down — but they’ll sneak in a trap number later (like x⁻²). Remember: for x⁻¹, you use ln|x|, not the power rule. Every year, that’s a 1-mark loss for hundreds of students.

🎯 Pattern check:
If all the powers are integers and there’s no product or bracket to chain through — it’s power rule, nothing fancy.
See a bracket? Get suspicious.

🔁 Example 2 — Substitution (OCR Loves These Hidden Ones)

Here’s one straight from OCR style papers:

∫ (2x)(x² + 1)³ dx

Looks messy, right? But notice that the derivative of x² + 1 is 2x — it’s literally sitting there waiting to be used. That’s your clue to substitute.

Let u = x² + 1, so du = 2x dx.
Substitute, and the integral becomes ∫ u³ du = u⁴/4 + C = (x² + 1)⁴/4 + C.

💬 Teacher aside:
“This one’s sneaky — it looks worse than it is. OCR likes to bury easy substitutions under intimidating brackets.”

🎯 Pattern check:
If there’s a bracket and the outside term looks like the derivative of the inside — go for substitution.

🔀 Example 3 — Integration by Parts (Edexcel’s Mid-Paper Curveball)

Now, this is where students usually freeze.
Let’s take:

∫ x eˣ dx

When you see a product (two different types of functions multiplied), think integration by parts.

Formula reminder:

∫ u dv = uv − ∫ v du

Pick u = x (because it simplifies when differentiated), and dv = eˣ dx (because eˣ stays the same when integrated).

Then:

du = 1 dx
v = eˣ

Plug in:
∫ x eˣ dx = x eˣ − ∫ eˣ dx = eˣ(x − 1) + C

💬 Teacher reflection:
I once told a student, “Choose u like you’d choose snacks for a long study session — pick the one that gets smaller when you differentiate.” He never forgot again.

🎯 Pattern check:
Product of x and another function? Think “by parts.”
If it’s trig × exponential or x × ln(x), same rule applies.

📘 Board tip:
Edexcel loves pairing this with exam timing stress. They’ll drop it in the middle of a paper where it looks like substitution might work — but it’s not. Trust your instinct on the pattern.

🌊 Example 4 — Trig Integrals (OCR and AQA Variations)

Okay, trig integrals.
They look terrifying, but most follow two patterns: either use identities first or spot the derivative relationship.

Example:

∫ sin(2x) dx

That’s easy once you remember: sin(2x) = 2 sin(x) cos(x).
So you can either use substitution (u = 2x → du = 2 dx) or halve it directly.

Result:

∫ sin(2x) dx = −½ cos(2x) + C

💬 Teacher aside:
“Always deal with the number in front of x first — that’s the bit everyone forgets to divide by.”

Common exam mistake? Students integrate sin(2x) like sin(x), forgetting the 2.
AQA loves this as a one-mark trap near the end of a question.

🎯 Pattern check:
If there’s a trig of something more complicated than x, you’ll probably need substitution.
If it’s multiplied by its own derivative, even better — it collapses neatly.

🧠 Example 5 — Definite Integrals & Real Context (Edexcel and OCR Mix)

Now the big one: real-world integration.

A particle moves with velocity v = 3t² + 2t. Find the displacement between t = 0 and t = 3.

Integration here isn’t just algebra — it’s physics in disguise.

You’re finding the area under the curve, so:

s = ∫₀³ (3t² + 2t) dt = [t³ + t²]₀³ = (27 + 9) − 0 = 36.

🎯 Pattern check:
Velocity → displacement
Acceleration → velocity
That’s it. OCR likes to phrase these as “find the total distance travelled” — careful, that sometimes means taking absolute values if direction changes.

💬 Mini anecdote:
I had a student panic mid-mock because he’d “forgotten which one was which.”
I just said, “Differentiate to go up, integrate to come back down.” He grinned, wrote it on the front of his paper, and got full marks.

🧭 Common Mistakes to Avoid

Let’s be honest — most lost marks in integration come from tiny slips, not big gaps.

Here are the classics:

Forgetting +C — AQA occasionally gives a 1-mark penalty just for this.
Mixing up substitution limits — When you change variable, change the limits or switch back before substituting.
Wrong choice of u or dv — In by-parts, choosing u poorly creates chaos.
Sign errors in trig — sin integrates to −cos, not cos. OCR loves catching this.

💬 Exam-board insight:
Edexcel papers often chain these traps: they’ll test substitution, limits, and sign in one question.
So slow down, label every line, and write logic like a teacher explaining it back.

⏱️ Integration Timing Strategy

If you’ve ever run out of time mid-paper, integration’s probably to blame.
It’s method-heavy — every step earns marks.

💬 Teacher advice:
Time each method once in practice:

Power rule → under 1 min
Substitution → ~2 min
By parts → 3–4 min
Definite integral → add 1 min

When you know these rhythms, you feel if a question’s going too long — and that alone saves entire grades.

📘 Mark Scheme Logic: How to Think Like an Examiner

Every board rewards structure. They’re not looking for magic answers — they want to see:

Substitution written clearly (u = …)
Limits changed correctly (if definite)
Each line logically built (no mental jumps)

💬 Teacher tip:
“Write your steps like someone else has to mark them — because they do.”
AQA mark schemes literally award marks for method visibility, even if the final number’s off.

OCR often says “award follow-through marks” — meaning your clear process can rescue partial credit even with small errors.

🔍 Spotting Which Method to Use (Summary Table)

Clue	Method	Example
Pure powers of x	Power rule	3x² + 4x + 7
Bracket + derivative outside	Substitution	(2x)(x² + 1)³
Two different terms multiplied	By parts	x eˣ
Trig with coefficient inside	Substitution	sin(2x)
Velocity / acceleration context	Definite	∫₀³ v(t) dt

💬 Teacher aside:
“I call this my ‘integration bingo card’. Spot two clues, and the method usually picks itself.”

🧩 Real Exam Application — Mixed Question (OCR, 2022)

∫₀^{π/2} x sin(x) dx

OCR loves this combo — it’s both trig and product. So, yes, integration by parts.

Let u = x → du = dx
dv = sin(x) dx → v = −cos(x)

∫ x sin(x) dx = −x cos(x) + ∫ cos(x) dx = −x cos(x) + sin(x)

Evaluate 0 to π/2:
[−(π/2)(0) + 1] − [0 + 0] = 1.

🎯 Exam insight: OCR expects the final exact value (not decimal).
Always write “= 1” — not “≈ 1.00”.

💬 Teacher Reflection

I’ll be honest — when I first started teaching, I used to overcomplicate integration.
Every example turned into a lecture.
Now, I tell my students: “Think patterns first, algebra second.”

Once that clicks, integration becomes relaxing. It’s like solving a puzzle where all the pieces fit once you know the shapes.

And if you ever blank in an exam?
Just breathe, look at the structure, and ask: “What am I looking through right now — a power, a product, or a bracket?”
That question alone unlocks 90% of integration problems.

🚀 Next Steps

If you’ve followed these five examples, you’ve covered almost every integration pattern that appears across AQA, Edexcel, and OCR.

Now it’s about practice with structure and timing.

Strengthen your skills with our A Level Maths Half-Term Revision Course — proven 3-day sessions designed to boost your marks fast.
Learn how examiners think, not just what to memorise.

Author Bio – S. Mahandru

S. Mahandru is Head of Maths at Exam.tips. With over 15 years of teaching experience, he simplifies algebra and provides clear examples and strategies to help GCSE students achieve their best.

🧭 Next topic:

“Now strengthen the reasoning skills needed for A Level proof questions.”

❓ Quick FAQs

How do I know whether to use substitution or by parts?

If it’s a product, try by parts.
If one term looks like the derivative of another, it’s substitution.

What if I forget the formula for by parts in the exam?

Derive it quickly: integrate the product rule backwards — or write “∫u dv = uv − ∫v du” in the formula booklet space before you start.

How many marks do I lose for forgetting +C?

Usually just one — but it can cost full credit on an indefinite question if consistency matters. Always add it.

Integration Made Simple: 5 Real A Level Questions Explained Step-by-Step