Fundamental Theorem of Calculus : A Level Maths Guide

The Fundamental Theorem of Calculus links differentiation and integration as inverse processes. It is one of the key ideas in A Level Maths integration and is essential for working with definite integrals.

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What Is the Fundamental Theorem of Calculus?

The Fundamental Theorem of Calculus (FTC) connects two major areas of calculus:

• Differentiation
• Integration

The theorem has two main parts.

FTC Part 1

If:

F(x) = ∫ₐˣ f(t) dt

then:

F′(x) = f(x)

In plain English:

Differentiating an integral with a variable upper limit gives back the original function.

FTC Part 2

If F(x) is an antiderivative of f(x), then:

∫ₐᵇ f(x) dx = F(b) − F(a)

In plain English:

To evaluate a definite integral, find an antiderivative, substitute the upper and lower limits, and subtract.

Together, these two results show that differentiation and integration undo each other.

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FTC Part 1 - Differentiating an Integral

The key rule is:

d/dx [∫ₐˣ f(t) dt] = f(x)

This means that if an integral has a variable upper limit x, differentiating the integral gives you the integrand evaluated at x.

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Worked Example 1

Find G′(x) if:

G(x) = ∫₀ˣ (t² + 3t) dt

Step 1

Identify the integrand:

f(t) = t² + 3t

Step 2

Apply FTC Part 1.

Replace t with x:

G′(x) = x² + 3x

Answer

G′(x) = x² + 3x

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Chain Rule Extension

Sometimes the upper limit is not simply x.

Instead, it may be another function such as:

u(x)

In this situation, the chain rule must also be used.

The rule becomes:

d/dx [∫ₐᵘ⁽ˣ⁾ f(t) dt] = f(u(x)) × u′(x)

So there are two steps:

1. Substitute the upper-limit function into the integrand.
2. Multiply by the derivative of the upper-limit function.

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Worked Example 2

Find:

d/dx [∫₀ˣ² cos(t) dt]

Step 1

Identify the integrand:

f(t) = cos(t)

and the upper-limit function:

u(x) = x²

Step 2

Differentiate the upper-limit function:

u′(x) = 2x

Step 3

Substitute x² into the integrand:

f(x²) = cos(x²)

Step 4

Multiply by u′(x):

cos(x²) × 2x

Answer

2x cos(x²)

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FTC Part 2 - Evaluating Definite Integrals

FTC Part 2 allows you to calculate the value of a definite integral.

The rule is:

∫ₐᵇ f(x) dx = F(b) − F(a)

where F(x) is an antiderivative of f(x).

The process is:

1. Integrate the function.
2. Substitute the upper limit.
3. Substitute the lower limit.
4. Calculate upper limit minus lower limit.

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Worked Example 3

Evaluate:

∫₁⁴ (2x + 1) dx

Step 1

Find the antiderivative:

F(x) = x² + x

Step 2

Substitute the upper limit:

F(4) = 4² + 4

F(4) = 16 + 4 = 20

Step 3

Substitute the lower limit:

F(1) = 1² + 1

F(1) = 1 + 1 = 2

Step 4

Subtract:

F(4) − F(1)

= 20 − 2

= 18

Answer

18

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Two Key Notes

1. Drop the +C

When working with indefinite integrals, you normally include the constant of integration:

F(x) + C

However, for a definite integral:

(F(b) + C) − (F(a) + C)

the constants cancel:

F(b) + C − F(a) − C

= F(b) − F(a)

Therefore:

Do not include +C when evaluating definite integrals.

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2. Definite Integrals Give Signed Area

A definite integral calculates the net signed area between a curve and the x-axis over an interval.

This means:

• Areas above the x-axis contribute positive values.
• Areas below the x-axis contribute negative values.
• Positive and negative areas can cancel each other.

Therefore, a definite integral can sometimes equal 0 even when there are regions between the curve and the x-axis.

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FTC Part 1 vs FTC Part 2

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Common Mistakes

1. Forgetting the Chain Rule

If the upper limit is:

u(x)

then:

d/dx [∫ₐᵘ⁽ˣ⁾ f(t) dt] = f(u(x)) × u′(x)

Do not write only:

f(u(x))

You must also multiply by:

u′(x)

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2. Subtracting in the Wrong Order

Always calculate:

F(b) − F(a)

That means:

Upper limit − Lower limit

not the other way around.

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3. Adding +C to a Definite Integral

For a definite integral, the constants cancel.

Therefore:

∫ₐᵇ f(x) dx = F(b) − F(a)

There is no +C in the final answer.

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4. Getting the Sign Wrong When Reversing Limits

Remember:

∫ₐᵇ f(x) dx = −∫ᵦᵃ f(x) dx

Reversing the limits changes the sign of the integral.

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5. Mixing Up FTC Part 1 and FTC Part 2

Ask yourself what the question requires.

If you are asked to:

Differentiate an integral

use FTC Part 1.

If you are asked to:

Evaluate a definite integral

use FTC Part 2.

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Frequently Asked Questions

Is the Fundamental Theorem of Calculus a GCSE Topic?

No.

The formal Fundamental Theorem of Calculus is an A Level Maths topic.

GCSE Maths introduces ideas involving gradients and areas, but the formal relationship between differentiation and integration is studied at a higher level.

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What Does the Fundamental Theorem of Calculus Tell Us?

The theorem tells us that differentiation and integration are inverse processes.

FTC Part 1 shows that differentiating an accumulated integral returns the original function.

FTC Part 2 shows that definite integrals can be evaluated using an antiderivative.

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What Is FTC Part 1?

FTC Part 1 states:

d/dx [∫ₐˣ f(t) dt] = f(x)

It means that differentiating an integral whose upper limit is x returns the original integrand.

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What Is FTC Part 2?

FTC Part 2 states:

∫ₐᵇ f(x) dx = F(b) − F(a)

where:

F′(x) = f(x)

It provides the standard method for evaluating definite integrals.

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Does +C Matter in a Definite Integral?

No.

The constants cancel:

(F(b) + C) − (F(a) + C)

= F(b) − F(a)

Therefore, do not include +C when evaluating a definite integral.

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What Does a Definite Integral Calculate?

A definite integral calculates the signed area between a curve and the x-axis from:

x = a

to:

x = b

Areas above the x-axis contribute positively, while areas below the x-axis contribute negatively.

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What Happens If the Upper Limit Is a Function of x?

If the upper limit is u(x) rather than simply x, combine the Fundamental Theorem of Calculus with the chain rule:

d/dx [∫ₐᵘ⁽ˣ⁾ f(t) dt] = f(u(x)) × u′(x)

For example:

d/dx [∫₀ˣ² cos(t) dt]

= cos(x²) × 2x

= 2x cos(x²)

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What Happens When the Limits of Integration Are Reversed?

Reversing the limits changes the sign:

∫ₐᵇ f(x) dx = −∫ᵦᵃ f(x) dx

For example:

∫₁⁴ f(x) dx = −∫₄¹ f(x) dx

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Key Rules to Remember

FTC Part 1

d/dx [∫ₐˣ f(t) dt] = f(x)

Chain Rule Extension

d/dx [∫ₐᵘ⁽ˣ⁾ f(t) dt] = f(u(x)) × u′(x)

FTC Part 2

∫ₐᵇ f(x) dx = F(b) − F(a)

where:

F′(x) = f(x)

Signed Area

• Area above the x-axis contributes positively.
• Area below the x-axis contributes negatively.

Constant of Integration

For definite integrals:

+C cancels

so it should not appear in the final answer.

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Quick Revision Table

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Final Summary

The Fundamental Theorem of Calculus provides the connection between differentiation and integration.

Remember these five key rules:

• FTC Part 1: d/dx [∫ₐˣ f(t) dt] = f(x)
• Chain rule extension: d/dx [∫ₐᵘ⁽ˣ⁾ f(t) dt] = f(u(x)) × u′(x)
• FTC Part 2: ∫ₐᵇ f(x) dx = F(b) − F(a)
• Signed area: regions below the x-axis contribute negative values
• +C cancels: do not include the constant of integration in a definite integral

These rules underpin A Level calculus and integration in A Level Maths and are essential for confidently working with definite integrals.