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Properties of
Definite Integrals

Master the key rules that let you split, combine, and simplify definite integrals: saving time and unlocking harder problems on the AP exam.

AP Calculus AB

What Is a Definite Integral?

A definite integral \(\displaystyle\int_a^b f(x)\,dx\) computes the net signed area between the curve \(y = f(x)\) and the \(x\)-axis from \(x = a\) to \(x = b\). Area above the axis is positive; area below is negative.

x y a b y = f(x)

Why Do These Properties Matter?

On the AP exam, you are often given individual integral values and asked to find a new integral built from them. Without these properties, you would need the antiderivative every time. With them, you can combine known values like puzzle pieces.

Efficiency
Avoid re-computing antiderivatives when values are already known.
Flexibility
Break a complex interval into simpler, familiar pieces.
Exam Ready
Multiple-choice and free-response questions test these rules directly.

The Big Five Properties

1
Zero-Width: \(\displaystyle\int_a^a f(x)\,dx = 0\)
2
Reverse Bounds: \(\displaystyle\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx\)
3
Constant Multiple: \(\displaystyle\int_a^b k\,f(x)\,dx = k\int_a^b f(x)\,dx\)
4
Sum/Difference: \(\displaystyle\int_a^b [f(x)\pm g(x)]\,dx = \int_a^b f(x)\,dx \pm \int_a^b g(x)\,dx\)
5
Splitting Interval: \(\displaystyle\int_a^b f(x)\,dx = \int_a^c f(x)\,dx + \int_c^b f(x)\,dx\)

Zero-Width & Reverse Bounds

a = b Zero Width Area = 0 a b Reverse Bounds Swap = negate

Constant Multiple & Sum Rule

Constant Multiple
$$\int_a^b k\,f(x)\,dx = k\int_a^b f(x)\,dx$$Pull any constant factor outside the integral.
Sum/Difference
$$\int_a^b [f(x)\pm g(x)]\,dx = \int_a^b f(x)\,dx \pm \int_a^b g(x)\,dx$$Split an integrand into separate integrals.

Splitting the Interval

a c b a to c c to b
\(\int_a^b f(x)\,dx = \color{#818cf8}{\int_a^c f(x)\,dx} + \color{#4ade80}{\int_c^b f(x)\,dx}\)

Linearity vs. Products

A critical distinction: the integral is linear (sums and constant multiples work), but it does not distribute over products or powers in any simple way.

Operation Valid? Rule
\(\int [f + g]\,dx\) ✓ Yes \(\int f\,dx + \int g\,dx\)
\(\int k\,f\,dx\) ✓ Yes \(k\int f\,dx\)
\(\int f \cdot g\,dx\) ✕ No \(\neq \int f\,dx \cdot \int g\,dx\)
\(\int [f]^n\,dx\) ✕ No \(\neq \left[\int f\,dx\right]^n\)

How to Apply the Properties

1
Identify what you know: List all given integral values and their bounds.
2
Match or adjust bounds: Use splitting or reversing to align intervals with known values.
3
Separate the integrand: Use the sum/difference rule to break apart combined functions.
4
Factor out constants: Pull any scalar multiples outside the integral sign.
5
Substitute and compute: Replace each integral with its known value and simplify.

Example 1: Sum Rule

1
Split the integrand: \(\displaystyle\int_1^4 3f(x)\,dx - \int_1^4 2g(x)\,dx\)
2
Factor out constants: \(3\displaystyle\int_1^4 f(x)\,dx - 2\int_1^4 g(x)\,dx\)
3
Substitute: \(3(6) - 2(-2) = 18 + 4 = \) 22

Example 2: Splitting Intervals

0 3 5 = 10 = 7 = ?

Example 3: Reverse Bounds + Splitting

1
Split \([2,6]\) at \(c=4\): \(\displaystyle\int_2^6 f(x)\,dx = \int_2^4 f(x)\,dx + \int_4^6 f(x)\,dx\)
2
Solve for \(\int_4^6\): \(8 = 3 + \int_4^6 f(x)\,dx \implies \int_4^6 f(x)\,dx = 5\)
3
Reverse bounds: \(\displaystyle\int_6^4 f(x)\,dx = -\int_4^6 f(x)\,dx = \) \(-5\)

Example 4: Combining All Properties

1
Split \(f\) on \([0,8]\): \(\int_3^8 f(x)\,dx = 12 - 5 = 7\)
2
Split \(g\) on \([0,8]\): \(\int_3^8 g(x)\,dx = -4 - 1 = -5\)
3
Combine: \(2(7) + 3(-5) = 14 - 15 = \) \(-1\)

Signed Area Matters

x
\(+\)
\(-\)
 a b

Common Mistakes to Avoid

✕ Distributing over products
Writing \(\int f \cdot g\,dx = \int f\,dx \cdot \int g\,dx\). The integral is not multiplicative.
✕ Forgetting the sign flip
When reversing bounds, forgetting to negate: \(\int_b^a f\,dx = -\int_a^b f\,dx\), not the same value.
✕ Pulling out variables
Writing \(\int x\,f(x)\,dx = x\int f(x)\,dx\). Only true constants can be factored out.

Properties Summary

Property Formula Key Detail
Zero-Width \(\int_a^a f\,dx = 0\) No interval, no area
Reverse Bounds \(\int_a^b f\,dx = -\int_b^a f\,dx\) Sign flips
Constant Multiple \(\int k\,f = k\int f\) Only constants, not variables
Sum/Difference \(\int(f\pm g)=\int f \pm \int g\) Linearity of integration
Splitting \(\int_a^b=\int_a^c+\int_c^b\) \(c\) can be anywhere

Question 1 of 5

If \(\displaystyle\int_1^5 f(x)\,dx = 9\) and \(\displaystyle\int_1^5 g(x)\,dx = -3\), what is \(\displaystyle\int_1^5 [4f(x) + 2g(x)]\,dx\)?

Question 2 of 5

If \(\displaystyle\int_0^7 f(x)\,dx = 15\) and \(\displaystyle\int_0^4 f(x)\,dx = 9\), what is \(\displaystyle\int_4^7 f(x)\,dx\)?

Question 3 of 5

What is \(\displaystyle\int_5^5 (3x^2 + \sin x)\,dx\)?

Question 4 of 5

If \(\displaystyle\int_2^8 f(x)\,dx = 10\), what is \(\displaystyle\int_8^2 f(x)\,dx\)?

Question 5 of 5

Which statement is always true?