11.6PureStretch

Integration by Parts

Integration by parts reverses the product rule. It is used to integrate a product such as x eˣ or x ln x, where one factor simplifies when differentiated.

28 min Video by Zeeshan Zamurred Integration

Edexcel A level Maths: 11.6 Integration By Parts (Part 1)Watch the full walkthrough before the notes below.

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What you'll be able to do

State the integration by parts formula
Choose u and dv/dx wisely
Apply to products like x eˣ, x ln x
Handle the special case ∫ln x dx

The formula

For a product, $\int u \frac{d v}{d x} d x = uv - \int v \frac{d u}{d x} d x$ . Choose $u$ to be the part that gets simpler when differentiated.

\int u \frac{d v}{d x} d x = uv - \int v \frac{d u}{d x} d x

Integration by parts.

u = x

\frac{d v}{d x} = e^{x}

;

\frac{d u}{d x} = 1

v = e^{x}

x e^{x} - \int e^{x} d x = x e^{x} - e^{x} + c

Answer

e^{x} (x - 1) + c

Tip — Choose u using LATE: Logs, Algebra, Trig, Exponentials — pick the earliest as u.

Integrating ln x

A neat trick: $\int ln x d x$ uses parts with $u = ln x$ and $\frac{d v}{d x} = 1$ , giving $x ln x - x + c$ .

Formula recap

\int u v^{'} d x = uv - \int v u^{'} d x

By parts.

\int ln x d x = x ln x - x + c

Special case.

Common mistakes to avoid

Choosing u as the part that gets more complicated.

Pick u so its derivative simplifies (use LATE order).

Dropping the minus sign before the second integral.

It is uv − ∫v u′ dx.

Key takeaways

∫u v′ dx = uv − ∫v u′ dx.
Choose u by LATE (logs first, exponentials last).
∫ln x dx = x ln x − x + c.

Test yourself

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