Integration by Parts (Pure Mathematics 3 – 9709) — Mathematics

Introduction

Integration by parts is one of the most important techniques in Pure Mathematics 3. It allows you to integrate products of functions that cannot be handled by standard forms or substitution alone. The syllabus specifically requires you to recognise when an integrand should be treated as a product, and to apply the technique confidently to expressions such as $x \sin 2x$ , $x^2 e^{-x}$ , $\ln x$ , and $x \tan^{-1} x$ . Questions involving this method appear regularly in Paper 3, often combined with definite integration or asking for an exact answer.

Core Concept

The method is derived from the product rule for differentiation. If $u$ and $v$ are functions of $x$ , the product rule states:

\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}

Rearranging and integrating both sides with respect to $x$ gives the integration by parts formula:

\int u \frac{dv}{dx}\, dx = uv - \int v \frac{du}{dx}\, dx

Choosing $u$ and $\dfrac{dv}{dx}$ is the critical skill. Use the LIATE priority order as a guide — whichever function type appears earliest in this list should be chosen as $u$ :

Priority	Type	Examples
1st	Logarithms	$\ln x$ , $\log_a x$
2nd	Inverse trig	$\tan^{-1} x$ , $\sin^{-1} x$
3rd	Algebraic (polynomial)	$x$ , $x^2$ , $x^3$
4th	Trigonometric	$\sin x$ , $\cos x$
5th	Exponential	$e^x$ , $e^{-x}$

The key principle: choose $u$ so that $\dfrac{du}{dx}$ is simpler than $u$ , and choose $\dfrac{dv}{dx}$ so that you can integrate it to find $v$ .

Special cases: For $\ln x$ and $\tan^{-1} x$ alone (apparently not a product), write the integrand as $1 \cdot \ln x$ or $1 \cdot \tan^{-1} x$ and set $u = \ln x$ (or $\tan^{-1} x$ ) and $\dfrac{dv}{dx} = 1$ .

Repeated application: For integrands like $x^2 e^{-x}$ , you must apply integration by parts twice, each time reducing the power of the polynomial factor.

Unlock the full Integration note with Nova

You're reading the preview. Unlock the complete note — every worked example, examiner pitfall and practice question — plus 24/7 AI tutoring from Nova that teaches directly from these notes.

Unlock with Nova Explore CAIE A-Level tutoring →

Introduction

Core Concept

Unlock the full Integration note with Nova

Keep learning