This is a beta course, so its structure, chapters, and examples may continue to change.

Linearity

Linearity is one of the most basic properties of definite integrals.

Basic Linearity Property

Theorem1

Let $f(x)$ and $g(x)$ be integrable on $[a, b]$ , and $k$ be a constant. Then:

Proof

Use linearity of sums and limits.

Example: Compute $\int_0^1 (2x^2 + 3x) \, dx$

Solution:

Use linearity to compute $\int_0^1 (3x^2 + 2x + 1) \, dx$ .

Reference Answer

linearitydefinite integral

Approach: Split the integrand into three separate terms and integrate each one on its own, then add the three results by linearity.

Steps:

Split: $\int_0^1 (3x^2 + 2x + 1) \, dx = 3\int_0^1 x^2 \, dx + 2\int_0^1 x \, dx + \int_0^1 1 \, dx$ .
First term: $3\int_0^1 x^2 \, dx = 3\left[\tfrac{x^3}{3}\right]_0^1 = 3\cdot\tfrac13 = 1$ .
Second term: $2\int_0^1 x \, dx = 2\left[\tfrac{x^2}{2}\right]_0^1 = 2\cdot\tfrac12 = 1$ .
Third term: $\int_0^1 1 \, dx = [x]_0^1 = 1$ .
Add the three results: $1 + 1 + 1 = 3$ .

Answer: $3$ .

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$\int$	mathematical symbol	integral	Definite integral sign
$k$	constant	constant	The constant multiplier in the linearity property
$f(x), g(x)$	function symbol	f of x, g of x	The two integrable functions whose integrals are combined
$[a, b]$	interval	closed interval	The interval on which both integrands are integrable
$x^2$	mathematical symbol	x squared	The power term integrated separately in the worked example
$x$	mathematical symbol	x	The linear term integrated separately in the worked example
$dx$	mathematical symbol	differential of x	The differential element of the integration variable $x$
$2, 3$	constant	two, three	The constant multipliers carried by the two terms

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