Parts Integration Method – Evaluating ∫ x eˣ dx

🧠 Parts Integration Method – Choosing u and dv

🧭 Why this integral is about choice, not the formula

When students first meet the integral $\int x e^x \, dx$ , it often feels deceptively simple. There’s only a product of two terms, and one of them — $e^x$ — integrates to itself. That can give the impression that the method is obvious.

In reality, this integral is testing something much quieter.

It is not asking whether you remember a formula. It is asking whether you can choose a sensible method and apply it deliberately, rather than mechanically. Students who rush into writing the integration by parts formula without thinking about the structure often make small errors that unravel the whole solution.

This kind of decision-making is a core part of A Level Maths problem-solving explained properly. The algebra only behaves once the choice does.

At this point in lessons, I usually pause and ask a simple question: what would happen if we differentiated the wrong part here?
That pause matters more than students expect.

This example follows directly from the general approach set out in Integration by Parts — Method & Exam Insight, particularly the structured choice of u and dv.

🔙 Previous topic:

Like optimisation questions involving maximum volume and fixed surface area, this relies on choosing the right method before doing any calculus.

📘 What the examiner is actually looking for

Examiners do not include $\int x e^x \, dx$ because it is difficult. They include it because it reveals how a student thinks when faced with a product that resists basic techniques.

Substitution does not naturally simplify the integral. Basic integration rules are not enough. Something else is required.

What the examiner wants to see is whether you can recognise that:

the integrand is a product,
one part becomes simpler when differentiated,
and the other does not become more complicated when integrated.

That recognition is the real skill being assessed.

🧠 The idea that everything rests on

Integration by parts is built around one guiding principle:

Differentiate what simplifies. Integrate what doesn’t get worse.

This principle matters far more than memorising a formula.

The standard formula is:

$\int u \, dv = uv – \int v \, du$

But this line is not the skill. It is just the tool. The skill lies in choosing $u$ and $dv$ so that the resulting integral is genuinely easier than the original one.

If the new integral looks just as awkward — or worse — then the choice was wrong, even if the formula was applied correctly.

✏️ Applying that idea to ∫ x eˣ dx

Consider the integral:

$\int x e^x \, dx$

Before writing anything down, it is worth slowing the thought process deliberately.

Differentiating $x$ simplifies it to $1$ .
Integrating $e^x$ leaves it unchanged.

That combination is ideal.

So we choose:

$u = x$
$dv = e^x \, dx$

From this, it follows that:

$du = dx$
$v = e^x$

This choice is not accidental. It is the result of thinking one step ahead.

This kind of deliberate setup is exactly what A Level Maths revision explained clearly tries to reinforce — method before mechanics.

🧮 Carrying out the method carefully

Now apply the integration by parts formula:

$\int x e^x \, dx = x e^x – \int e^x \, dx$

At this stage, notice how the new integral is immediately simpler than the original one. That is the sign that the method has been chosen well.

Integrating $e^x$ gives:

$\int e^x \, dx = e^x$

So the full result becomes:

$x e^x - e^x + C$

It is often helpful to factorise the final expression:

$e^x(x - 1) + C$

Both forms are completely acceptable.

🔍 A quiet examiner check students often skip

It is worth pausing here and noticing something important. The solution did not require speed, clever tricks, or heavy algebra. It required a calm choice at the start.

This is deliberate.

Integration by parts questions are designed so that the calculus itself is almost routine once the choice is correct. Examiners are testing whether you can plan before calculating.

That is why rushed setups often score fewer marks than slower, clearer ones.

🧠 Why this choice works (and others don’t)

Some students try choosing $u = e^x$ instead. Technically, that is allowed. But it leads to an integral that looks almost identical to the original one, which defeats the purpose of the method.

Examiners are not impressed by the fact that integration by parts was used. They are impressed by whether it was used well.

This distinction matters throughout the topic.

🔍 Where students usually lose marks

Most errors in this question are not mathematical in the traditional sense. They come from process issues.

Common problems include:

choosing $u$ without thinking ahead,
forgetting the minus sign in the formula,
failing to simplify the final expression,
or rushing straight into the formula without stating the choices clearly.

Integration by parts rewards clarity more than speed.

🧮 Worked Exam Example (Clearly Separated)

🧪 Worked Exam Example
Evaluate $\int x e^x \, dx$ .

Choose:

$u = x$ , $dv = e^x \, dx$

Then:

$du = dx$ , $v = e^x$

Apply the formula:

$\int x e^x \, dx = x e^x – \int e^x \, dx$

So:

$= x e^x - e^x + C$

Or equivalently:

$= e^x(x - 1) + C$

🧮 Why this integral matters later

This example is not included in exams by accident. It acts as a gateway to more complex integration by parts problems involving logarithms, inverse trigonometric functions, and repeated applications.

Students who are confident with this structure tend to stay much calmer when the algebra becomes heavier later on.

If this method still feels uncertain under pressure, structured support such as a A Level Maths Revision Course for real exam skill can help reinforce the decision-making behind the technique, not just the formula.

Author Bio – S. Mahandru

When students say integration by parts feels random, it’s usually because they’ve been taught the formula without the thinking behind it. In lessons, I spend more time discussing why a choice works than carrying out the algebra — because once the choice is right, the mechanics usually behave.

🧭 Next topic:

Once this standard method is secure, the tabular method becomes a natural next step for handling repeated integration by parts more efficiently in exams.

🎯 Final exam takeaway

If there is one habit to take from this question, it is this: think about the effect of differentiation and integration before writing the formula. Building that habit consistently — turns integration by parts from a stressful technique into a predictable one.

❓ Quick FAQs

🧭 How do I know when integration by parts is the right method?

A strong signal is the presence of a product where one part simplifies when differentiated and the other does not become more complicated when integrated. If differentiation makes one part cleaner and integration leaves the other manageable, integration by parts is usually worth considering. Examiners expect hesitation here — what they reward is a sensible decision, not instant certainty.

🧠 Does the order of choosing u and dv really matter?

Yes, very much. The formula itself does not change, but the resulting integral can be dramatically different. A poor choice can lead to an integral that is just as difficult as the original one. A good choice leads to immediate simplification. Examiners look carefully at this setup because it reveals understanding rather than memorisation.

⚖️ What if my final answer looks different from the mark scheme?

That is very common with integration by parts. Algebraic expressions can look different while being mathematically equivalent. Examiners are trained to recognise valid alternative forms. What matters is whether your working is correct and whether differentiating your answer would return the original integrand. Clear method protects marks even when final forms differ.