Combining Integrals with Equal or Contiguous Ranges

Updated August 2025

This lesson explains how to combine multiple integrals into a single expression by using the properties of equal ranges and contiguous intervals. You will learn to simplify complex calculus problems by merging integrands or extending the range of integration. These techniques are essential for solving multi step TMUA integration questions efficiently.

Core concept

Integration is a linear operation that allows functions with the same limits to be combined, $\int_{a}^{b} f(x) dx + \int_{a}^{b} g(x) dx = \int_{a}^{b} [f(x) + g(x)] dx$ , and contiguous intervals of the same function to be joined, $\int_{a}^{b} f(x) dx + \int_{b}^{c} f(x) dx = \int_{a}^{c} f(x) dx$ .

Combining Integrals with Equal Ranges

One of the most useful properties of integration is its linearity. If you have two separate integrals that share exactly the same lower and upper limits, you can combine them into one. This is essentially the reverse process of term by term integration. By summing or subtracting the integrands first, you may often find an expression that is far simpler to integrate than the two original parts.

As specified in the official guide, for any two functions $f(x)$ and $g(x)$ over the same range from 2 to 5, we have:

$\int_{2}^{5} f(x) dx + \int_{2}^{5} g(x) dx = \int_{2}^{5} [f(x) + g(x)] dx$

This rule applies whether you are dealing with additions or subtractions, provided the limits remain identical for both terms.

Combining Integrals with Contiguous Ranges

Integration can also be combined when the ranges are contiguous. This means that the upper limit of one integral is the same as the lower limit of the next. If the function being integrated, $f(x)$ , is the same in both integrals, you can merge them into a single integral that covers the total range.

Consider the following example from the examiner guide:

$\int_{2}^{4} f(x) dx + \int_{4}^{3} f(x) dx = \int_{2}^{3} f(x) dx$

Notice here that the limits do not have to be in increasing order for this property to hold. We can use the Fundamental Theorem of Calculus to justify why this works. If we let $F(x)$ be the antiderivative of $f(x)$ , then $F'(x) = f(x)$ . The sum of the integrals becomes:

$[F(4) - F(2)] + [F(3) - F(4)] = F(3) - F(2) = \int_{2}^{3} f(x) dx$

Unpacking the Logic of Merging Ranges

To develop a deeper formal understanding, we can look at the middle steps of the contiguous range example provided above. By breaking down the interval from 2 to 4 and using the property that swapping limits introduces a minus sign, we can see the cancellation clearly:

$\int_{2}^{4} f(x) dx + \int_{4}^{3} f(x) dx = \int_{2}^{3} f(x) dx + \int_{3}^{4} f(x) dx - \int_{3}^{4} f(x) dx = \int_{2}^{3} f(x) dx$

In this sequence, we have rewritten $\int_{2}^{4} f(x) dx$ as the sum $\int_{2}^{3} f(x) dx + \int_{3}^{4} f(x) dx$ , and we have used the fact that $\int_{4}^{3} f(x) dx = -\int_{3}^{4} f(x) dx$ . The two $\int_{3}^{4}$ terms cancel each other out, leaving only the desired result. You should ensure you have both an intuitive grasp and a formal algebraic understanding of why these combinations are valid regardless of the numerical values of the limits.

Key takeaways

Integrals with equal limits can be combined by adding or subtracting their integrands into a single function.
Contiguous ranges occur when the upper limit of one integral matches the lower limit of another for the same function.
The property $\int_{a}^{b} f(x) dx + \int_{b}^{c} f(x) dx = \int_{a}^{c} f(x) dx$ remains true even if the limits are not in numerical order.
Swapping the upper and lower limits of a definite integral always multiplies the value of the integral by negative one.

Tips

In the TMUA, always look at the limits before you start integrating. If you see multiple integrals of the same function, see if you can 'chain' them together. This often eliminates the need to evaluate the antiderivative multiple times, saving you valuable time.

Cautions

Be extremely careful with signs when combining integrals. A common error is forgetting that $\int_{b}^{a} f(x) dx$ is the negative of $\int_{a}^{b} f(x) dx$ . Always double check the direction of your contiguous limits.

Insight

This property is a direct consequence of the Fundamental Theorem of Calculus. Since $\int_{a}^{b} f(x) dx$ represents the total change in the antiderivative $F(x)$ from $a$ to $b$ , adding the change from $b$ to $c$ naturally gives the total change from $a$ to $c$ .

Worked Examples

Example 1

Let f be a polynomial with real coefficients.
The integral

I_{p,q}

where

p < q

is defined by

I_{p,q} = \int_{p}^{q} (f(x))^2 - (f(|x|))^2 dx

Which of the following statements must be true?
1

I_{p,q} = 0

only if

0 < p

f'(x) < 0

for all x only if

I_{p,q} < 0

for all

p < q < 0

I_{p,q} > 0

only if

p < 0

A:none of them

B:1 only

C:2 only

D:3 only

E:1 and 2 only

F:1 and 3 only

G:2 and 3 only

H:1, 2 and 3

Practice Questions

Practice Question 1

The function

f

is such that

f(0) = 0

, and

xf(x) > 0

for all non-zero values of

x

.
It is given that

\int_{-2}^{2} f(x)\,dx = 4

and

\int_{-2}^{2} |f(x)|\, dx = 8

Evaluate

\int_{-2}^{0} f(|x|)\,dx

-8

-6

-4

-2

2

4

6

8

Frequently asked questions

Can I combine integrals if the functions are different but the limits are contiguous?

No. To use the contiguous range property, the function $f(x)$ inside the integral must be identical. If the functions are different, the areas do not merge into a single continuous calculation of one function.

Does the value $b$ have to be between $a$ and $c$ for the range property to work?

No. The property $\int_{a}^{b} + \int_{b}^{c} = \int_{a}^{c}$ is algebraically valid for any real numbers $a$ , $b$ , and $c$ , provided the function is defined over the entire interval.

What happens if I am subtracting two contiguous integrals?

You can use the limit swapping rule to turn the subtraction into an addition. For example, $\int_{a}^{c} f(x) dx - \int_{b}^{c} f(x) dx = \int_{a}^{c} f(x) dx + \int_{c}^{b} f(x) dx$ , which then combines to $\int_{a}^{b} f(x) dx$ .

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