Introduction to Integration (Fundamental Theorem of Calculus) — Edexcel A-level Mathematics (Pure)

SPEC 8.1 • INTEGRATION

Introduction to Integration

Understanding the Fundamental Theorem of Calculus and the reverse of differentiation

What is Integration?

Integration is the reverse process of differentiation.

Function $y = f (x)$

Differentiate

Derivative $\frac{d y}{d x}$

Integrate

Function $y = f (x)$

Key Concept

If we know the derivative (gradient function) of a curve, integration allows us to find the original function.

The Fundamental Theorem of Calculus

Fundamental Theorem

Integration is the reverse process of differentiation.

If \frac{d y}{d x} = f (x), then y = \int f (x) d x

This means:

Differentiation and integration are inverse operations

One "undoes" the other

We can verify any integration by differentiating the result

Tip

Think of it like addition and subtraction, or multiplication and division — opposite operations that cancel each other out.

Integral Notation

Notation

\int f (x) d x

SymbolMeaning

\int

The integral sign — tells us to integrate

f (x)

The integrand — the function we are integrating

d x

Indicates we are integrating with respect to

x

Reading the Notation

\int 3 x^{2} d x

is read as "the integral of

3 x^{2}

with respect to

x

"

The Constant of Integration

When we differentiate a constant, it disappears:

\frac{d}{d x} (x^{2} + 5) = 2 x \frac{d}{d x} (x^{2} - 3) = 2 x \frac{d}{d x} (x^{2} + 100) = 2 x

All three functions give the same derivative! So when we integrate

2 x

, we cannot know what constant was there originally.

Rule

When integrating, we must always add a constant of integration, written as

+ c

\int 2 x d x = x^{2} + c

Important

Forgetting

+ c

is the most common error in integration. Always include it for indefinite integrals!

Family of Curves

The constant

c

means integration gives us a family of curves, not just one curve.

If

\frac{d y}{d x} = 2 x

, then:

y = x^{2} + 0

(when

c = 0

)

y = x^{2} + 2

(when

c = 2

)

y = x^{2} - 3

(when

c = - 3

)

These are all vertical translations of each other.

Visualising

All curves in the family have the same gradient at any given

x

-value, but are shifted up or down.

Finding a Specific Curve

To find one specific curve from the family, we need extra information — usually a point that the curve passes through.

Key Points Summary

The Fundamental Theorem

Integration is the reverse of differentiation. If

\frac{d y}{d x} = f (x)

, then

y = \int f (x) d x

Notation

\int f (x) d x

\int

— integral sign

f (x)

— integrand

d x

— with respect to

x

Constant of Integration

Always add

+ c

to indefinite integrals

This represents an unknown constant lost during differentiation

Verification Method

Always check your integration by differentiating the answer — you should get back to the original integrand.

Summary

What You've Learned

ConceptKey Point

Fundamental TheoremIntegration is the reverse of differentiation

Notation

\int f (x) d x

means "integrate f(x) with respect to x"

Constant of IntegrationAlways add

+ c

to indefinite integrals

Finding Specific CurvesUse a given point to find the value of

c

VerificationDifferentiate your answer to check it's correct

Looking Ahead

In the next lesson, we'll learn the rules for integrating powers of

x

, giving us a systematic method rather than relying on reversing differentiation.