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Integration by Parts: When (and When Not)
🧮Integration by Parts — When (and When Not)
Okay, quick question before we start — do you remember the product rule from differentiation?
Good. Because integration by parts is basically that rule, but in reverse.
It’s the maths equivalent of rewinding a song: same structure, just backwards.
Every student I’ve ever taught has looked at the formula and thought, “That looks horrible.”
And honestly? Fair reaction. It does look ugly on paper.
But once you see what it’s doing, it’s actually quite logical — and kind of neat.
🔙 Previous topic:
“Revisit common differentiation pitfalls before moving on to integration by parts.”
🧭 Why the Exam Boards Love It
This technique shows up everywhere — AQA, Edexcel, OCR — because it forces you to think.
They like it because it checks more than one skill at once: algebra, differentiation, and judgement.
AQA might throw it in with trig.
Edexcel likes it inside a 6-mark proof.
OCR sometimes makes you use it after substitution.
So yeah, it’s worth knowing inside out.
⚙️ What It Actually Means
Let’s start where it comes from. You know the product rule:
= u'v + uv'
Now, integrate both sides with respect to x.
\int (u'v + uv'),dx = uv
Split the integrals:
\int u'v,dx + \int uv',dx = uv
Then rearrange it. Move one of those integrals across.
\int u,dv = uv – \int v,du
And that’s it — the whole “monster” formula comes straight from the product rule.
So, nothing mysterious, just the same idea flipped around.
But — and this is where most people go wrong — you have to choose u and dv carefully.
That choice decides whether your next line gets easier… or a whole lot worse.
📏 How to Choose u and dv
Right, so here’s the rule most teachers use: LIATE.
It’s short for Logarithmic, Inverse trig, Algebraic, Trig, Exponential.
That’s your priority list for choosing u.
So if you’ve got a mix, like x e^x, go down the list.
Algebraic (x) comes before exponential (e^x), so set u = x, dv = e^x dx.
Differentiate u → du = dx.
Integrate dv → v = e^x.
Now substitute into the formula.
\int x e^x,dx = x e^x – \int e^x,dx
And because ∫e^x dx = e^x, your answer is:
x e^x – e^x + C
Done. Smooth. No drama.
🧠 Little teaching aside: I tell my students to say “LIATE” out loud before they start — it stops the guessing game. The rhythm helps you choose without freezing.
❗ Trap #1 – Picking Them the Wrong Way Around
Let’s mess it up deliberately.
Say you swap them. Let u = e^x, dv = x dx.
Then du = e^x dx and v = ½x².
Now plug that in:
\int e^x x,dx = \tfrac{1}{2}x^2 e^x – \int \tfrac{1}{2}x^2 e^x,dx
You can already see the problem — the new integral looks nastier than before.
So here’s the fix: if the result looks worse, you’ve probably chosen u and dv the wrong way round.
Swap them and try again.
⚙️ Example – When Trig Joins the Party
Now, things get interesting when trigs appear. Try this:
\int e^x \cos x,dx
Hmm. Both e^x and cos x look equally nice, right?
So just pick one. Let’s take u = e^x, dv = cos x dx.
Differentiate u → du = e^x dx.
Integrate dv → v = sin x.
Now plug in:
\int e^x \cos x,dx = e^x \sin x – \int e^x \sin x,dx
Okay, that new integral looks suspiciously familiar.
It’s another exponential–trig combo.
So call the original integral I:
I = e^x \sin x – \int e^x \sin x,dx
Now do integration by parts again on that last bit (swap cos for sin).
Eventually, you’ll end up back with I and can solve algebraically.
It feels like witchcraft the first time, but it’s neat once you see the loop close.
🧠 Quick warning from experience: any time exponentials and trigs show up together, expect to use parts twice. Keep calm; it works out nicely in the end.
📘 When You Shouldn’t Use It
Okay, this is where good students waste marks.
Integration by parts is powerful — but not universal.
If one part differentiates nicely and the other integrates easily, great.
If not, it’s probably a trap.
Example:
\int x^2 e^{x^2},dx
Looks tempting, doesn’t it? You might think: “There’s an x² and an e^{x²} — perfect!”
Nope. If you try it, you’ll quickly find you can’t integrate e^{x²}. It’s not solvable in standard form.
So what do you do? Substitution instead.
Let t = x² → dt = 2x dx.
Then it becomes:
\tfrac{1}{2}\int e^t,dt = \tfrac{1}{2}e^t + C = \tfrac{1}{2}e^{x^2} + C
That’s your cue: if the integral gets worse after setting it up, abandon ship.
❗ Don’t force integration by parts where substitution fits better.
⚙️ Example – The Famous Logarithm One
Every exam board loves this one.
\int x \ln x,dx
Okay, so by LIATE, logarithmic functions go first.
So set u = ln x, dv = x dx.
Differentiate u → du = (1/x) dx.
Integrate dv → v = ½x².
Plug those in:
\int x \ln x,dx = \tfrac{1}{2}x^2 \ln x – \int \tfrac{1}{2}x^2 \cdot \tfrac{1}{x},dx
Simplify that:
\tfrac{1}{2}x^2 \ln x – \tfrac{1}{2}\int x,dx = \tfrac{1}{2}x^2 \ln x – \tfrac{1}{4}x^2 + C
And there’s your answer. Clean and examiner-approved.
🧠 Teacher’s check: did u get simpler when you differentiated it?
If yes, you picked correctly. That’s your internal compass.
❗ Trap #2 – Forgetting the Minus
Every year, someone loses marks because they drop that minus in the formula.
Write it clearly somewhere on your page:
\int u,dv = uv – \int v,du
Minus, not plus. It’s the biggest single-sign loss of marks in this whole topic.
I tell my classes: “Plus for product, minus for parts.”
Say it, write it, circle it — whatever helps it stick.
📏 When the Method Loops
Sometimes, doing integration by parts brings you back to where you started.
Take:
\int e^x \sin x,dx
You do one round, and the next integral becomes e^x cos x.
Do it again, and you’re back at e^x sin x.
When that happens, label the whole thing I, bring all I terms to one side, and solve for I.
You’ll get a tidy expression in the end.
✅ Quick mental note: when you see sine or cosine with e^x, expect the circle to repeat once.
🧠 Quick Teacher Story
A few years back, one of my students — let’s call him Dan — decided to use integration by parts on everything.
He even tried it on \int x^3,dx.
He spent five lines on something that takes one step.
We laughed, but it stuck with him forever: don’t use a cannon to swat a fly.
If a method feels like overkill, it probably is.
📘 Real-World Link
You might wonder, “When will I ever need this outside class?”
Fair question.
Integration by parts shows up in physics — waves, damping, electrical circuits — basically anytime something’s changing while something else changes.
Like exponential decay combined with oscillation.
For example, \int e^{-x}\sin x,dx describes a decaying wave.
Engineers, data scientists, and even economists use it when modelling systems that fade or repeat over time.
So yes — it matters far beyond your exam paper.
✅ Quick Recap Table
Concept | Key Idea |
Formula | \int u,dv = uv – \int v,du |
Choice rule | Use LIATE to decide u |
Use it when | One part gets simpler when differentiated |
Avoid when | Neither part simplifies or the result gets uglier |
Common mistakes | Wrong sign, wrong order, forgetting to simplify |
Looping cases | e^x\sin x or e^x\cos x types |
🚀 What to Do Next
If this still feels like juggling, that’s fine. You’re supposed to practise slowly before it clicks.
Want a structure to fit these questions into your week? Check out How to Revise for A Level Maths Effectively — it walks you through a plan that balances practice and rest.
Try these three tonight:
1️⃣ \int x e^x,dx
2️⃣ \int x \ln x,dx
3️⃣ \int e^x \sin x,dx
And this time, don’t rush. Talk yourself through the steps.
When you start explaining the rule out loud, you’ll find you remember it twice as fast.
If you’d like more of these step-by-step teacher breakdowns — substitution, partial fractions, trig tricks — they’re all covered in our A Level Maths Half-Term Revision Course, with diagrams and past-paper examples for AQA, Edexcel, and OCR.
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:
Revisit core integration methods to see how they lead naturally into parts.
💬 FAQs
What’s the quickest way to spot when I need the chain rule?
If you see brackets — that’s your cue.
Anything like (2x + 3)^5 , \sin(3x^2) , or e^{4x} has one function sitting inside another. That’s when the chain rule applies.
A quick trick: say to yourself, “outside and inside.” Differentiate the outer bit, then multiply by the derivative of the inner one. Do both, every time — no guessing needed.
How can I stop missing terms when differentiating long expressions?
Slow down and read each term out loud.
It sounds silly, but it works — your voice forces your brain to notice what your eyes skip.
For example, when differentiating y = 3x^3 + 5x – 7 , say “three x cubed, plus five x, minus seven.”
That rhythm helps you remember to handle every term — even the constants.
Why do I lose marks even when my differentiation looks right?
Usually because you’ve stopped too early.
You might have the correct derivative, but if you don’t simplify or tidy it, examiners can’t always see the logic.
Always expand brackets, collect like terms, and present your final answer neatly.
A clear line of reasoning shows understanding — and that’s where the marks live.