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Properties of
Definite Integrals

Master the algebraic rules that let you split, combine, and simplify definite integrals: essential tools for AP Calculus BC.

AP Calculus BC

What Is a Definite Integral?

The definite integral \(\displaystyle\int_a^b f(x)\,dx\) represents the 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) signed area

Why Properties Matter

Many AP problems give you individual integral values and ask you to compute a related integral. Instead of antidifferentiating, you apply algebraic properties to combine known values.

Simplify
Break complex integrands into known pieces
Combine
Merge or split intervals to match given data
Evaluate
Compute new integrals without antiderivatives

The Core Properties

1
Constant Multiple: \(\displaystyle\int_a^b c\,f(x)\,dx = c\int_a^b f(x)\,dx\)
2
Sum / Difference: \(\displaystyle\int_a^b [f(x) \pm g(x)]\,dx = \int_a^b f(x)\,dx \pm \int_a^b g(x)\,dx\)
3
Additivity of Intervals: \(\displaystyle\int_a^b f(x)\,dx + \int_b^c f(x)\,dx = \int_a^c f(x)\,dx\)
4
Reversal of Limits: \(\displaystyle\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx\)
5
Zero-Width Interval: \(\displaystyle\int_a^a f(x)\,dx = 0\)

Additivity of Intervals

x y a b c Area 1 Area 2

Reversal of Limits

a b +A a → b : positive b a \u2212A b → a : negative =

Linearity of Integration

The constant-multiple and sum/difference rules combine into one powerful idea: the definite integral is a linear operator.

This means you can pull out constants and split sums term by term: exactly the same rules you already use for derivatives.

Linear vs Non-linear

Valid (Linear) Invalid (Non-linear)
Sum \(\int [f+g] = \int f + \int g\) \(\int f \cdot g = \int f \cdot \int g\)
Constant \(\int c\,f = c\int f\) \(\int [f]^2 = [\int f]^2\)
Intervals \(\int_a^b + \int_b^c = \int_a^c\) \(\int_a^b \cdot \int_b^c = \int_a^c\)

Integral of a Constant

y = k a b
\(b - a\)
 k

How to Apply the Properties

1
Identify given values. List every \(\int_a^b f(x)\,dx\) and \(\int_a^b g(x)\,dx\) that you know.
2
Match the target interval. If limits don't match, use additivity or reversal to adjust.
3
Decompose the integrand. Use linearity to break the expression into known integrals.
4
Substitute and compute. Plug in the known values and simplify.

Example 1: Linearity

1
Split by linearity: \(\displaystyle 4\int_1^5 f(x)\,dx - 2\int_1^5 g(x)\,dx\)
2
Substitute: \(4(7) - 2(3)\)
3
Compute: \(28 - 6 = \) \(22\)

Example 2: Additivity

1
Additivity: \(\displaystyle\int_0^2 f(x)\,dx + \int_2^6 f(x)\,dx = \int_0^6 f(x)\,dx\)
2
Solve: \(\displaystyle\int_2^6 f(x)\,dx = 10 - 4\)
3
Result: \(6\)
0 2 6 4 ?

Example 3: Combined Properties

1
Additivity on \(f\): \(\displaystyle\int_1^7 f(x)\,dx = 8 + (-3) = 5\)
2
Linearity: \(\displaystyle 3\int_1^7 f(x)\,dx + \int_1^7 g(x)\,dx\)
3
Substitute: \(3(5) + 5 = 15 + 5 = \) \(20\)

Example 4: Limit Reversal

1
Reversal: \(\displaystyle\int_8^3 f(x)\,dx = -\int_3^8 f(x)\,dx = -12\)
2
Zero-width: \(\displaystyle\int_3^3 f(x)\,dx = 0\)
3
Total: \(-12 + 0 = \) \(-12\)

Common Mistakes

✕ Distributing Products
Writing \(\int f \cdot g = \int f \cdot \int g\). Integration is linear, not multiplicative.
✕ Ignoring Sign on Reversal
Treating \(\int_b^a f = \int_a^b f\). Flipping limits flips the sign.
✕ Mismatched Intervals
Adding integrals over different functions on different intervals without checking overlap.

Properties Summary

Property Formula When to Use
Constant Multiple \(c\int f\) Factor out scalars
Sum / Difference \(\int f \pm \int g\) Split mixed integrands
Additivity \(\int_a^b + \int_b^c = \int_a^c\) Bridge intervals
Reversal \(-\int_b^a\) Flip wrong-way limits
Zero-Width \(0\) Same upper and lower limit
Constant Integrand \(k(b-a)\) Rectangle area

Question 1 of 5

If \(\displaystyle\int_2^6 f(x)\,dx = 9\), what is \(\displaystyle\int_6^2 f(x)\,dx\)?

Question 2 of 5

Given \(\displaystyle\int_0^5 f(x)\,dx = 12\) and \(\displaystyle\int_0^5 g(x)\,dx = 4\), find \(\displaystyle\int_0^5 [2f(x) - 3g(x)]\,dx\).

Question 3 of 5

If \(\displaystyle\int_1^4 f(x)\,dx = 7\) and \(\displaystyle\int_4^9 f(x)\,dx = 5\), what is \(\displaystyle\int_1^9 f(x)\,dx\)?

Question 4 of 5

Which of the following is always true?

Question 5 of 5

Given \(\displaystyle\int_0^{10} f(x)\,dx = 20\), \(\displaystyle\int_0^4 f(x)\,dx = 6\), \(\displaystyle\int_0^{10} g(x)\,dx = -2\), and \(\displaystyle\int_0^4 g(x)\,dx = 0\), find: $$\int_4^{10} [f(x) - g(x)]\,dx$$