Basic Integral Properties and Rules

Math Calculus • Integrals

Written by STEM Calculators Team Published June 15, 2026 Updated June 15, 2026

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Theory

What are integral properties?

Integral properties are rules that let us rewrite integrals without directly finding an antiderivative. They are especially useful when the integrand contains symbolic functions such as \(f(x)\), \(g(x)\), or \(h(x)\).

For example:

\[ \begin{aligned} \int_a^b\left(2f(x)+3g(x)\right)\,dx &= 2\int_a^b f(x)\,dx + 3\int_a^b g(x)\,dx \end{aligned} \]

The main idea is that integrals behave nicely with addition, subtraction, and constant multiples.

Linearity of integrals

The most important property is linearity.

\[ \begin{aligned} \int_a^b\left(\alpha f(x)+\beta g(x)\right)\,dx &= \alpha\int_a^b f(x)\,dx + \beta\int_a^b g(x)\,dx \end{aligned} \]

Here, \(\alpha\) and \(\beta\) are constants. The same rule works for more than two terms.

For example:

\[ \begin{aligned} \int_a^b\left(4f(x)-h(x)+5\right)\,dx &= 4\int_a^b f(x)\,dx - \int_a^b h(x)\,dx + \int_a^b 5\,dx \end{aligned} \]

Constant multiple rule

A constant factor can be pulled outside the integral.

\[ \begin{aligned} \int_a^b kf(x)\,dx &= k\int_a^b f(x)\,dx \end{aligned} \]

Example:

\[ \begin{aligned} \int_a^b 7f(x)\,dx &= 7\int_a^b f(x)\,dx \end{aligned} \]

The bounds do not change. Only the constant moves outside.

Sum and difference rules

The integral of a sum is the sum of the integrals:

\[ \begin{aligned} \int_a^b\left(f(x)+g(x)\right)\,dx &= \int_a^b f(x)\,dx + \int_a^b g(x)\,dx \end{aligned} \]

The integral of a difference is the difference of the integrals:

\[ \begin{aligned} \int_a^b\left(f(x)-g(x)\right)\,dx &= \int_a^b f(x)\,dx - \int_a^b g(x)\,dx \end{aligned} \]

These two rules combine with the constant multiple rule to form linearity.

Worked example: \(\int_a^b [2f(x)+3g(x)]\,dx\)

Start with:

\[ \begin{aligned} \int_a^b\left(2f(x)+3g(x)\right)\,dx \end{aligned} \]

Split the sum:

\[ \begin{aligned} \int_a^b\left(2f(x)+3g(x)\right)\,dx &= \int_a^b 2f(x)\,dx + \int_a^b 3g(x)\,dx \end{aligned} \]

Pull out the constants:

\[ \begin{aligned} \int_a^b 2f(x)\,dx + \int_a^b 3g(x)\,dx &= 2\int_a^b f(x)\,dx + 3\int_a^b g(x)\,dx \end{aligned} \]

Therefore:

\[ \begin{aligned} \int_a^b\left(2f(x)+3g(x)\right)\,dx &= 2\int_a^b f(x)\,dx + 3\int_a^b g(x)\,dx \end{aligned} \]

Constant terms inside definite integrals

A constant term can also be integrated over an interval.

\[ \begin{aligned} \int_a^b k\,dx &= k(b-a) \end{aligned} \]

Example:

\[ \begin{aligned} \int_a^b\left(f(x)+5\right)\,dx &= \int_a^b f(x)\,dx + \int_a^b 5\,dx\\ &= \int_a^b f(x)\,dx + 5(b-a) \end{aligned} \]

Additivity over intervals

If \(c\) is between \(a\) and \(b\), then:

\[ \begin{aligned} \int_a^b f(x)\,dx &= \int_a^c f(x)\,dx + \int_c^b f(x)\,dx \end{aligned} \]

This works because the total signed area from \(a\) to \(b\) is the sum of the signed area from \(a\) to \(c\) and the signed area from \(c\) to \(b\).

The same property applies to a whole expression:

\[ \begin{aligned} \int_a^b\left(2f(x)+3g(x)\right)\,dx &= \int_a^c\left(2f(x)+3g(x)\right)\,dx + \int_c^b\left(2f(x)+3g(x)\right)\,dx \end{aligned} \]

Reversing bounds

Reversing the bounds changes the sign of the integral.

\[ \begin{aligned} \int_a^b f(x)\,dx &= -\int_b^a f(x)\,dx \end{aligned} \]

This is because moving from \(a\) to \(b\) has the opposite orientation from moving from \(b\) to \(a\).

Example:

\[ \begin{aligned} \int_2^5 f(x)\,dx &= -\int_5^2 f(x)\,dx \end{aligned} \]

Zero-width intervals

If the lower and upper bounds are the same, the interval has no width.

\[ \begin{aligned} \int_a^a f(x)\,dx &= 0 \end{aligned} \]

This is true for any function \(f(x)\) that is defined at the point.

Even and odd symmetry

Symmetry can simplify integrals over intervals of the form \([-r,r]\).

If \(f(x)\) is even, then \(f(-x)=f(x)\), and:

\[ \begin{aligned} \int_{-r}^{r} f(x)\,dx &= 2\int_0^r f(x)\,dx \end{aligned} \]

If \(f(x)\) is odd, then \(f(-x)=-f(x)\), and:

\[ \begin{aligned} \int_{-r}^{r} f(x)\,dx &= 0 \end{aligned} \]

Odd symmetry cancels signed areas on the left and right sides of the origin.

Comparison property

If one function is always below another function on an interval, their integrals follow the same order.

If:

\[ \begin{aligned} f(x)\le g(x) \qquad\text{for all }x\in[a,b], \end{aligned} \]

then:

\[ \begin{aligned} \int_a^b f(x)\,dx &\le \int_a^b g(x)\,dx \end{aligned} \]

This property is often used to estimate integrals without finding exact values.

Common mistakes

Dropping the bounds: every split definite integral must keep the correct lower and upper bounds.
Forgetting constants: \(\int_a^b 2f(x)\,dx=2\int_a^b f(x)\,dx\), not just \(\int_a^b f(x)\,dx\).
Changing signs incorrectly: reversing bounds changes the sign of the entire integral.
Forgetting constant terms: \(\int_a^b 5\,dx=5(b-a)\).
Using symmetry without checking it: even and odd rules only work if the function actually has the stated symmetry and the interval is symmetric.
Confusing additivity with linearity: additivity splits the interval; linearity splits the integrand.

Frequently Asked Questions

What is the linearity property of integrals?

Linearity says that the integral of a linear combination equals the same linear combination of the integrals.

Can constants be pulled outside an integral?

Yes. The constant multiple rule says that ∫ kf(x) dx = k∫ f(x) dx.

Do the bounds stay the same when splitting an integral?

Yes. When you split ∫ from a to b of 2f(x)+3g(x), each new integral keeps the same bounds a and b.

What is additivity over intervals?

Additivity says that ∫ from a to b of f(x) dx equals ∫ from a to c of f(x) dx plus ∫ from c to b of f(x) dx.

What happens when bounds are reversed?

Reversing bounds changes the sign: ∫ from a to b of f(x) dx = -∫ from b to a of f(x) dx.

What is the value of an integral from a to a?

It is zero because the interval has no width.

How does odd symmetry help?

If a function is odd, then its integral over a symmetric interval [-r,r] is 0.

How does even symmetry help?

If a function is even, then its integral over [-r,r] equals twice its integral over [0,r].

Can this calculator find the actual antiderivative of f(x)?

No. This calculator applies general integral properties to symbolic functions. It does not need the explicit formula for f(x).

When can the calculator give a numeric result?

It gives a numeric result when all required base integral values are entered and constant-term contributions can be evaluated from the bounds.

Basic Integral Properties and Rules

Integral expression

Optional known integral values

Output and graph settings

Quick examples

Integral properties graph

Step-by-step property justification

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Frequently Asked Questions

Integral expression

Optional known integral values

Output and graph settings

Quick examples

Integral properties graph

Step-by-step property justification

Rate this calculator

Frequently Asked Questions

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