Integration

Q: How should you choose the function $u$ when using integration by parts?

Use the ILATE mnemonic: Inverse Trig, Logarithmic, Algebraic, Trigonometric, Exponential. Choose $u$ based on which type comes first in the list.

Pure Mathematics (YPM01) · Pearson Edexcel International A Level · 10 min read · Updated: 6 Sept 2026

Welcome to Unit P3: The Integration Chapter!

Hello future mathematicians! Integration is often seen as the biggest hurdle in Pure Mathematics 3, but don't worry. If differentiation is the process of finding the slope (rate of change), then integration is the process of putting everything back together – it’s the reverse of differentiation!

In this chapter, we move beyond simple power rules and learn powerful techniques—like substitution, breaking up fractions, and parts—that allow us to integrate almost anything thrown our way. Mastering these methods is crucial not just for your exams, but because integration is the tool we use to find areas, volumes, and solve real-world problems in physics and engineering.

1. Essential Prerequisite: The Standard Integrals and the Chain Rule in Reverse

Before diving into the advanced methods, let’s quickly refresh the P2 concepts that form the bedrock of P3.

1.1 The Constant of Integration (+C)

Always, always, always remember the Constant of Integration, $+C$, for indefinite integrals.

Why? Because when you differentiate a constant, it becomes zero. Therefore, when you integrate, you lose information about the original constant. $+C$ represents that missing constant.
Common Mistake: Forgetting the $+C$. This is a simple lost mark!

1.2 Integrating Standard Functions (P2 Review)

You must know these immediately. They are the keys to all other integration methods:

$\int e^x \, dx = e^x + C$
$\int \frac{1}{x} \, dx = \ln|x| + C$ (Remember the modulus sign!)
$\int \cos x \, dx = \sin x + C$
$\int \sin x \, dx = -\cos x + C$

1.3 Integrating Functions of the Form $f(ax+b)$

This is the simplest form of the "Chain Rule in Reverse" and is used constantly in P3:

If you know $\int f(x) \, dx = F(x) + C$, then:
$$\int f(ax+b) \, dx = \frac{1}{a} F(ax+b) + C$$

Example: To integrate $\cos(3x+5)$, we know $\int \cos x \, dx = \sin x$.
Therefore, $\int \cos(3x+5) \, dx = \frac{1}{3} \sin(3x+5) + C$.

Key Takeaway for Section 1

Always check if the integral is indefinite (+C) or definite (has limits). Know your standard integrals and how to handle linear factors inside functions.

2. Integration by Substitution (Changing the Variable)

Integration by Substitution is perhaps the most powerful tool. It's used when the integrand (the function you are integrating) involves a function and its derivative multiplied together, or when the expression is just too complicated to handle directly.

Analogy: Imagine trying to read a technical manual written in highly complex Latin. Substitution is like finding the key code (the substitution $u$) that instantly translates the manual into simple English.

2.1 The Two Clues for Substitution

Look closely at the expression. Substitution works best when:

There is a complex term raised to a power, or inside a root/logarithm.
You see a function AND its derivative multiplied together (or divided).

If you see $\int f(g(x)) \cdot g'(x) \, dx$, then let $u = g(x)$.

2.2 Step-by-Step Process for Substitution

Choose $u$: Select the complex part of the function (usually the part inside the bracket, root, or denominator) and set it equal to $u$.
Find $du$: Differentiate $u$ with respect to $x$: $\frac{du}{dx}$.
Find $dx$: Rearrange the differential equation to make $dx$ the subject.
Example: If $\frac{du}{dx} = 2x$, then $dx = \frac{du}{2x}$.
Substitute: Replace every term in the original integral (including $dx$) with terms involving $u$ and $du$. The goal is for all $x$ terms to cancel out completely!
Integrate: Solve the resulting, usually much simpler, integral with respect to $u$.
Substitute Back (Indefinite Integral): Replace $u$ with your original expression in $x$. Don’t forget $+C$!

2.3 Handling Definite Integrals (The Limits Trick)

If you have limits ($a$ and $b$) on your integral, you have a choice:

Option A (Harder): Ignore the limits, solve the integral, substitute back to $x$, and then apply the original limits.
Option B (Recommended!): Convert the limits from $x$-values to $u$-values in Step 3.

If $u = g(x)$, substitute the original upper limit $x=b$ into $g(x)$ to find the new upper limit, and substitute $x=a$ to find the new lower limit.

This means you integrate, evaluate using the new $u$-limits, and you never have to substitute back to $x$. Time saver!

Quick Review: The Special Case $\int \frac{f'(x)}{f(x)} \, dx$

This is the direct result of substitution, but is so common it's worth memorizing:

$$\int \frac{f'(x)}{f(x)} \, dx = \ln|f(x)| + C$$

Tip: If the numerator is the derivative of the denominator, the answer is the natural logarithm of the denominator.

Key Takeaway for Section 2

Substitution is a system of translation. If you choose $u$ correctly (often the complex part), everything should simplify and the variable $x$ must disappear entirely before integration.

3. Integration by Parts (Reversing the Product Rule)

We use integration by parts when we need to integrate a product of two different types of functions—for example, a polynomial times an exponential, or an algebraic term times a logarithm.

3.1 The Formula

This formula is derived by reversing the differentiation product rule and must be used precisely:
$$\int u \frac{dv}{dx} \, dx = uv - \int v \frac{du}{dx} \, dx$$

The Goal: We swap a difficult integral ($\int u \frac{dv}{dx} \, dx$) for a new, hopefully easier, integral ($\int v \frac{du}{dx} \, dx$).

3.2 Choosing $u$ and $\frac{dv}{dx}$

The success of integration by parts depends entirely on which function you choose as $u$ and which you choose as $\frac{dv}{dx}$.

$u$ (The Differentiator): This function needs to become simpler when differentiated.
$\frac{dv}{dx}$ (The Integrator): This function needs to be easy to integrate.

Use the mnemonic ILATE to help you choose $u$:
I - Inverse Trigonometric functions (rarely seen in P3)
L - Logarithmic functions ($\ln x$)
A - Algebraic functions ($x^n, x$)
T - Trigonometric functions ($\sin x, \cos x$)
E - Exponential functions ($e^x$)

You should choose $u$ as the function that comes first in the ILATE list.

Example: To integrate $\int x e^x \, dx$:
$x$ is Algebraic (A), $e^x$ is Exponential (E). A comes before E in ILATE.
Therefore, choose $u = x$ and $\frac{dv}{dx} = e^x$.

3.3 Step-by-Step Process for Parts

Identify & Calculate the Four Parts:
- Choose $u$ (Differentiate to find $\frac{du}{dx}$).
- Choose $\frac{dv}{dx}$ (Integrate to find $v$).
Apply the Formula: Plug the four parts into the formula: $uv - \int v \frac{du}{dx} \, dx$.
Solve the New Integral: Integrate the simplified integral $\int v \frac{du}{dx} \, dx$.

3.4 The Case of Repeated Integration by Parts (Recurrence)

Sometimes, after the first application of the formula, the new integral ($\int v \frac{du}{dx} \, dx$) still requires integration by parts (e.g., integrating $\int x^2 e^x \, dx$). You must apply the formula a second time.

A related case involves integrals that recur, such as $\int e^x \cos x \, dx$. After two applications of the formula, the original integral might reappear on the right side. When this happens:

Let $I = \int e^x \cos x \, dx$. The equation becomes $I = \text{stuff} - I$.
You simply rearrange to solve for $I$: $2I = \text{stuff}$, so $I = \frac{1}{2} \text{stuff}$.

Key Takeaway for Section 3

Integration by Parts is all about trading one integral for another. Use ILATE to correctly choose $u$ as the function that gets simpler when differentiated (usually Logarithms or Algebraic terms).

4. Integrating Rational Functions (Using Partial Fractions)

Rational functions are fractions where the numerator and denominator are both polynomials (e.g., $\frac{2x+3}{x^2+x}$). We cannot integrate these directly, but we can break them down into simpler pieces using Partial Fractions.

4.1 The Method Overview

The trick is to use the algebraic method of partial fractions (which you learned earlier in P3) to rewrite the complex fraction as a sum of simpler fractions.

Example: Instead of integrating $\int \frac{3x+1}{(x-1)(x+2)} \, dx$, we decompose it into: $$\int \left( \frac{A}{x-1} + \frac{B}{x+2} \right) \, dx$$ These simpler fractions integrate directly into logarithm functions using the formula from Section 2.3.

4.2 Types of Factors and Integration Result

Since P3 integration of rational functions relies heavily on the $\int \frac{f'(x)}{f(x)} \, dx$ rule, all resulting partial fractions will integrate to a logarithm.

| Type of Factor | Partial Fraction Form | Integral Result | |---|---|---| | Linear Factor $(ax+b)$ | $\frac{A}{ax+b}$ | $\frac{A}{a}\ln|ax+b|$ | | Repeated Factor $(ax+b)^2$ | $\frac{A}{ax+b} + \frac{B}{(ax+b)^2}$ | $\frac{A}{a}\ln|ax+b| - \frac{B}{a(ax+b)}$ |

Note on Repeated Factors: The term with the squared denominator, $\frac{B}{(ax+b)^2}$, is integrated using the power rule (rewrite it as $B(ax+b)^{-2}$).

4.3 Handling Improper Fractions

If the degree of the numerator is equal to or greater than the degree of the denominator (e.g., $\frac{x^2+1}{x^2+x}$), it is an improper fraction.

Crucial Step: You must perform algebraic long division (or synthetic division) first.
The result will be: $$\text{Improper Fraction} = \text{Quotient} + \frac{\text{Remainder}}{\text{Original Denominator}}$$
You integrate the Quotient (usually a simple polynomial) directly, and then apply partial fractions to the remaining proper fraction.

Key Takeaway for Section 4

If you see a complex fraction, think Partial Fractions. If the degree of the numerator is high, think Long Division FIRST. The resulting simple fractions should almost always integrate to $\ln$ terms.

5. Integrating Trigonometric Functions using Identities

Many trigonometric expressions cannot be integrated directly. Instead, we use standard trigonometric identities to rewrite the expression into terms we know how to integrate (e.g., $\cos(ax)$ or $\sin(bx)$).

5.1 Handling Squared Terms ($\sin^2 x$ and $\cos^2 x$)

You cannot integrate $\sin^2 x$ or $\cos^2 x$ using simple power rules or substitution. We must use the double angle identities for $\cos 2x$:

Identity 1: $\cos 2x = 2\cos^2 x - 1$
Rearranging gives: $$\cos^2 x = \frac{1}{2}(1 + \cos 2x)$$
Identity 2: $\cos 2x = 1 - 2\sin^2 x$
Rearranging gives: $$\sin^2 x = \frac{1}{2}(1 - \cos 2x)$$

Once rewritten, these expressions integrate easily using the rule for $f(ax+b)$ (Section 1.3).

5.2 Handling Products of Sines and Cosines ($\sin A \cos B$, etc.)

To integrate products like $\int \sin 5x \cos 2x \, dx$, we must use the Product-to-Sum Identities (sometimes provided in the formula booklet):

$2 \sin A \cos B = \sin(A+B) + \sin(A-B)$
$2 \cos A \cos B = \cos(A+B) + \cos(A-B)$
$2 \sin A \sin B = \cos(A-B) - \cos(A+B)$

The product is replaced by a sum, which we then integrate term by term.

5.3 Handling Odd Powers (e.g., $\sin^3 x$)

When you encounter odd powers, substitution usually works brilliantly.

Example: To integrate $\int \sin^3 x \, dx$:

Separate one factor: $\int \sin^2 x \cdot \sin x \, dx$
Use the Pythagorean Identity on the squared term: $\int (1 - \cos^2 x) \sin x \, dx$
Let $u = \cos x$. Then $\frac{du}{dx} = -\sin x$, so $\sin x \, dx = -du$.
Substitute: $\int (1 - u^2) (-du) = \int (u^2 - 1) \, du$
Integrate the simple polynomial in $u$.

5.4 Integrating $\tan x$ and $\cot x$

These are solved using the $\int \frac{f'(x)}{f(x)} \, dx$ rule (substitution is assumed):

$\int \tan x \, dx = \int \frac{\sin x}{\cos x} \, dx$
(Let $f(x) = \cos x$, then $f'(x) = -\sin x$. We need a minus sign in the numerator.) $$\int \tan x \, dx = -\ln|\cos x| + C$$ (Or $\ln|\sec x| + C$)
$\int \cot x \, dx = \ln|\sin x| + C$

Key Takeaway for Section 5

If a trigonometric integral looks hard, your first thought should be: "Which identity can simplify this?" Use double angle identities for squares, product-to-sum identities for products, and substitution for odd powers.

Chapter Summary and Final Encouragement

Congratulations! You have covered the four key pillars of advanced integration in P3.

The Integration Checklist:

Can I integrate it directly or using the $f(ax+b)$ rule? (If yes, done!)
Is there a function and its derivative present? (Try Substitution).
Is it a product of two different function types (e.g., $x$ and $\ln x$)? (Try Parts - ILATE).
Is it a complex fraction? (Try Partial Fractions).
Is it a tricky trig function ($\sin^2 x$, product)? (Try Identities).

Integration takes practice. Don't worry if the right method doesn't jump out immediately. Try one technique, and if it makes the integral more complicated, stop, wipe the slate clean, and try the next one. You've got this!

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

Why is the constant of integration $+C$ required for indefinite integrals?

It accounts for the constant term that becomes zero when a function is differentiated, as we lose information about it during the process.

What is the integral of $\frac{1}{x}$ with respect to $x$?

$\int \frac{1}{x} \, dx = \ln|x| + C$.

How do you integrate a function of the form $f(ax+b)$?

$\int f(ax+b) \, dx = \frac{1}{a} F(ax+b) + C$, where $F$ is the integral of $f$.

What are the two main indicators that integration by substitution should be used?

Substitution is used when there is a complex term raised to a power (or inside a root) or when a function and its derivative are both present in the integrand.

What is the recommended method for handling limits in definite integrals during substitution?

Convert the original $x$-limits into $u$-limits using the substitution formula so that you do not have to substitute back to $x$ at the end.

What is the result of the special case integral $\int \frac{f'(x)}{f(x)} \, dx$?

$\int \frac{f'(x)}{f(x)} \, dx = \ln|f(x)| + C$.

What is the formula for integration by parts?

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

How should you choose the function $u$ when using integration by parts?

Use the ILATE mnemonic: Inverse Trig, Logarithmic, Algebraic, Trigonometric, Exponential. Choose $u$ based on which type comes first in the list.

What should you do if the original integral reappears during integration by parts?

Rearrange the equation to group the identical integrals on one side and solve for the integral algebraically.

How do you integrate a complex rational function like $\frac{3x+1}{(x-1)(x+2)}$?

Decompose the fraction into partial fractions first, then integrate the resulting simpler terms into logarithms.

What is the first step when the numerator of a rational function has a degree equal to or higher than the denominator?

Perform algebraic long division to turn the improper fraction into a polynomial plus a proper fraction.

Which identity is used to integrate $\sin^2 x$?

$\sin^2 x = \frac{1}{2}(1 - \cos 2x)$.

How do you integrate a product of sines and cosines, such as $\sin 5x \cos 2x$?

Use the Product-to-Sum identities to rewrite the product as a sum of two separate trigonometric functions.

What is the integral of $\tan x$ and how is it derived?

$\int \tan x \, dx = \ln|\sec x| + C$. It is derived by rewriting $\tan x$ as $\frac{\sin x}{\cos x}$ and using the $\frac{f'(x)}{f(x)}$ rule.

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