Properties of definite integrals worksheet

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Sigma Prep · sigmaprep.io/worksheets Name Properties of Definite Integrals Date Period Use the properties of definite integrals. Given $\int_{- 1}^{1} f (x) d x = 9$ , $\int_{1}^{6} f (x) d x = - 6$ and $\int_{- 1}^{1} g (x) d x = 4$ , find $\int_{- 1}^{6} f (x) d x$ Given $\int_{0}^{3} f (x) d x = - 6$ and $\int_{0}^{3} g (x) d x = - 5$ , find $\int_{0}^{3} (3 f (x) - 2 g (x)) d x$ Given $\int_{- 1}^{6} f (x) d x = - 3$ and $\int_{2}^{6} f (x) d x = 7$ , find $\int_{- 1}^{2} - 4 f (x) d x$ $f$ is even and $\int_{- 3}^{3} f (x) d x = 24$ . Find $\int_{0}^{3} f (x) d x$ Given $\int_{0}^{1} f (x) d x = - 3$ , $\int_{1}^{5} f (x) d x = - 7$ and $\int_{0}^{1} g (x) d x = 2$ , find $\int_{5}^{1} f (x) d x$ Given $\int_{- 1}^{5} f (x) d x = 5$ and $\int_{- 1}^{5} g (x) d x = - 9$ , find $\int_{- 1}^{5} (5 f (x) + 6) d x$ Given $\int_{1}^{6} f (x) d x = - 5$ and $\int_{2}^{6} f (x) d x = - 5$ , find $\int_{1}^{2} f (x) d x$ $f$ is odd and $\int_{0}^{5} f (x) d x = 4$ . Find $\int_{- 5}^{0} f (x) d x$ Given $\int_{- 2}^{3} f (x) d x = - 6$ , $\int_{3}^{5} f (x) d x = 3$ and $\int_{- 2}^{3} g (x) d x = - 2$ , find $\int_{- 2}^{3} [- 2 f (x) - g (x)] d x$ Given $\int_{1}^{6} f (x) d x = 1$ and $\int_{1}^{6} g (x) d x = - 8$ , find $\int_{6}^{1} (2 f (x) + 3 g (x)) d x$ Given $\int_{- 1}^{1} f (x) d x = 3$ and $\int_{0}^{1} f (x) d x = - 8$ , find $\int_{0}^{- 1} f (x) d x$ $f$ is even, $g$ is odd, $\int_{0}^{1} f (x) d x = - 1$ and $\int_{0}^{1} g (x) d x = 6$ . Find $\int_{- 1}^{1} (3 f (x) + 4 g (x)) d x$ Given $\int_{- 3}^{3} f (x) d x = - 8$ , $\int_{3}^{7} f (x) d x = 6$ and $\int_{- 3}^{3} g (x) d x = - 1$ , find $\int_{- 3}^{7} f (x) d x$ Given $\int_{1}^{5} - 2 f (x) d x = 18$ , find $\int_{1}^{5} f (x) d x$ Given $\int_{0}^{4} f (x) d x = - 3$ and $\int_{3}^{4} f (x) d x = 7$ , find $\int_{0}^{3} f (x) d x$ Use symmetry to find $\int_{- 5}^{5} (4 x^{3} - 5 x - 6) d x$ Given $\int_{0}^{1} f (x) d x = - 3$ , $\int_{1}^{5} f (x) d x = 1$ and $\int_{0}^{1} g (x) d x = - 9$ , find $\int_{5}^{1} f (x) d x$ Given $\int_{- 2}^{2} f (x) d x = - 8$ and $\int_{- 2}^{2} g (x) d x = - 1$ , find $\int_{- 2}^{2} (- 3 f (x) + 5) d x$ Given $\int_{- 3}^{3} f (x) d x = - 12$ and $\int_{1}^{3} f (x) d x = 5$ , find $\int_{- 3}^{1} 3 f (x) d x$ $f$ is odd, $g$ is even, $\int_{0}^{4} f (x) d x = - 6$ and $\int_{0}^{4} g (x) d x = 3$ . Find $\int_{- 4}^{0} (- 3 f (x) - 3 g (x)) d x$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Properties of Definite Integrals

Date Period

Use the properties of definite integrals.

Given $\int_{- 1}^{1} f (x) d x = 9$ , $\int_{1}^{6} f (x) d x = - 6$ and $\int_{- 1}^{1} g (x) d x = 4$ , find $\int_{- 1}^{6} f (x) d x$
Given $\int_{0}^{3} f (x) d x = - 6$ and $\int_{0}^{3} g (x) d x = - 5$ , find $\int_{0}^{3} (3 f (x) - 2 g (x)) d x$
Given $\int_{- 1}^{6} f (x) d x = - 3$ and $\int_{2}^{6} f (x) d x = 7$ , find $\int_{- 1}^{2} - 4 f (x) d x$
$f$ is even and $\int_{- 3}^{3} f (x) d x = 24$ . Find $\int_{0}^{3} f (x) d x$
Given $\int_{0}^{1} f (x) d x = - 3$ , $\int_{1}^{5} f (x) d x = - 7$ and $\int_{0}^{1} g (x) d x = 2$ , find $\int_{5}^{1} f (x) d x$
Given $\int_{- 1}^{5} f (x) d x = 5$ and $\int_{- 1}^{5} g (x) d x = - 9$ , find $\int_{- 1}^{5} (5 f (x) + 6) d x$
Given $\int_{1}^{6} f (x) d x = - 5$ and $\int_{2}^{6} f (x) d x = - 5$ , find $\int_{1}^{2} f (x) d x$
$f$ is odd and $\int_{0}^{5} f (x) d x = 4$ . Find $\int_{- 5}^{0} f (x) d x$
Given $\int_{- 2}^{3} f (x) d x = - 6$ , $\int_{3}^{5} f (x) d x = 3$ and $\int_{- 2}^{3} g (x) d x = - 2$ , find $\int_{- 2}^{3} [- 2 f (x) - g (x)] d x$
Given $\int_{1}^{6} f (x) d x = 1$ and $\int_{1}^{6} g (x) d x = - 8$ , find $\int_{6}^{1} (2 f (x) + 3 g (x)) d x$
Given $\int_{- 1}^{1} f (x) d x = 3$ and $\int_{0}^{1} f (x) d x = - 8$ , find $\int_{0}^{- 1} f (x) d x$
$f$ is even, $g$ is odd, $\int_{0}^{1} f (x) d x = - 1$ and $\int_{0}^{1} g (x) d x = 6$ . Find $\int_{- 1}^{1} (3 f (x) + 4 g (x)) d x$
Given $\int_{- 3}^{3} f (x) d x = - 8$ , $\int_{3}^{7} f (x) d x = 6$ and $\int_{- 3}^{3} g (x) d x = - 1$ , find $\int_{- 3}^{7} f (x) d x$
Given $\int_{1}^{5} - 2 f (x) d x = 18$ , find $\int_{1}^{5} f (x) d x$
Given $\int_{0}^{4} f (x) d x = - 3$ and $\int_{3}^{4} f (x) d x = 7$ , find $\int_{0}^{3} f (x) d x$
Use symmetry to find $\int_{- 5}^{5} (4 x^{3} - 5 x - 6) d x$
Given $\int_{0}^{1} f (x) d x = - 3$ , $\int_{1}^{5} f (x) d x = 1$ and $\int_{0}^{1} g (x) d x = - 9$ , find $\int_{5}^{1} f (x) d x$
Given $\int_{- 2}^{2} f (x) d x = - 8$ and $\int_{- 2}^{2} g (x) d x = - 1$ , find $\int_{- 2}^{2} (- 3 f (x) + 5) d x$
Given $\int_{- 3}^{3} f (x) d x = - 12$ and $\int_{1}^{3} f (x) d x = 5$ , find $\int_{- 3}^{1} 3 f (x) d x$
$f$ is odd, $g$ is even, $\int_{0}^{4} f (x) d x = - 6$ and $\int_{0}^{4} g (x) d x = 3$ . Find $\int_{- 4}^{0} (- 3 f (x) - 3 g (x)) d x$



Sigma Prep · sigmaprep.io/worksheets Name Properties of Definite IntegralsAnswer keyVersion 1 Date Period Use the properties of definite integrals. Given $\int_{- 1}^{1} f (x) d x = 9$ , $\int_{1}^{6} f (x) d x = - 6$ and $\int_{- 1}^{1} g (x) d x = 4$ , find $\int_{- 1}^{6} f (x) d x$ $3$ Given $\int_{0}^{3} f (x) d x = - 6$ and $\int_{0}^{3} g (x) d x = - 5$ , find $\int_{0}^{3} (3 f (x) - 2 g (x)) d x$ $- 8$ Given $\int_{- 1}^{6} f (x) d x = - 3$ and $\int_{2}^{6} f (x) d x = 7$ , find $\int_{- 1}^{2} - 4 f (x) d x$ $40$ $f$ is even and $\int_{- 3}^{3} f (x) d x = 24$ . Find $\int_{0}^{3} f (x) d x$ $12$ Given $\int_{0}^{1} f (x) d x = - 3$ , $\int_{1}^{5} f (x) d x = - 7$ and $\int_{0}^{1} g (x) d x = 2$ , find $\int_{5}^{1} f (x) d x$ $7$ Given $\int_{- 1}^{5} f (x) d x = 5$ and $\int_{- 1}^{5} g (x) d x = - 9$ , find $\int_{- 1}^{5} (5 f (x) + 6) d x$ $61$ Given $\int_{1}^{6} f (x) d x = - 5$ and $\int_{2}^{6} f (x) d x = - 5$ , find $\int_{1}^{2} f (x) d x$ $0$ $f$ is odd and $\int_{0}^{5} f (x) d x = 4$ . Find $\int_{- 5}^{0} f (x) d x$ $- 4$ Given $\int_{- 2}^{3} f (x) d x = - 6$ , $\int_{3}^{5} f (x) d x = 3$ and $\int_{- 2}^{3} g (x) d x = - 2$ , find $\int_{- 2}^{3} [- 2 f (x) - g (x)] d x$ $14$ Given $\int_{1}^{6} f (x) d x = 1$ and $\int_{1}^{6} g (x) d x = - 8$ , find $\int_{6}^{1} (2 f (x) + 3 g (x)) d x$ $22$ Given $\int_{- 1}^{1} f (x) d x = 3$ and $\int_{0}^{1} f (x) d x = - 8$ , find $\int_{0}^{- 1} f (x) d x$ $- 11$ $f$ is even, $g$ is odd, $\int_{0}^{1} f (x) d x = - 1$ and $\int_{0}^{1} g (x) d x = 6$ . Find $\int_{- 1}^{1} (3 f (x) + 4 g (x)) d x$ $- 6$ Given $\int_{- 3}^{3} f (x) d x = - 8$ , $\int_{3}^{7} f (x) d x = 6$ and $\int_{- 3}^{3} g (x) d x = - 1$ , find $\int_{- 3}^{7} f (x) d x$ $- 2$ Given $\int_{1}^{5} - 2 f (x) d x = 18$ , find $\int_{1}^{5} f (x) d x$ $- 9$ Given $\int_{0}^{4} f (x) d x = - 3$ and $\int_{3}^{4} f (x) d x = 7$ , find $\int_{0}^{3} f (x) d x$ $- 10$ Use symmetry to find $\int_{- 5}^{5} (4 x^{3} - 5 x - 6) d x$ $- 60$ Given $\int_{0}^{1} f (x) d x = - 3$ , $\int_{1}^{5} f (x) d x = 1$ and $\int_{0}^{1} g (x) d x = - 9$ , find $\int_{5}^{1} f (x) d x$ $- 1$ Given $\int_{- 2}^{2} f (x) d x = - 8$ and $\int_{- 2}^{2} g (x) d x = - 1$ , find $\int_{- 2}^{2} (- 3 f (x) + 5) d x$ $44$ Given $\int_{- 3}^{3} f (x) d x = - 12$ and $\int_{1}^{3} f (x) d x = 5$ , find $\int_{- 3}^{1} 3 f (x) d x$ $- 51$ $f$ is odd, $g$ is even, $\int_{0}^{4} f (x) d x = - 6$ and $\int_{0}^{4} g (x) d x = 3$ . Find $\int_{- 4}^{0} (- 3 f (x) - 3 g (x)) d x$ $- 27$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Properties of Definite IntegralsAnswer keyVersion 1

Date Period

Use the properties of definite integrals.

Given $\int_{- 1}^{1} f (x) d x = 9$ , $\int_{1}^{6} f (x) d x = - 6$ and $\int_{- 1}^{1} g (x) d x = 4$ , find $\int_{- 1}^{6} f (x) d x$ $3$
Given $\int_{0}^{3} f (x) d x = - 6$ and $\int_{0}^{3} g (x) d x = - 5$ , find $\int_{0}^{3} (3 f (x) - 2 g (x)) d x$ $- 8$
Given $\int_{- 1}^{6} f (x) d x = - 3$ and $\int_{2}^{6} f (x) d x = 7$ , find $\int_{- 1}^{2} - 4 f (x) d x$ $40$
$f$ is even and $\int_{- 3}^{3} f (x) d x = 24$ . Find $\int_{0}^{3} f (x) d x$ $12$
Given $\int_{0}^{1} f (x) d x = - 3$ , $\int_{1}^{5} f (x) d x = - 7$ and $\int_{0}^{1} g (x) d x = 2$ , find $\int_{5}^{1} f (x) d x$ $7$
Given $\int_{- 1}^{5} f (x) d x = 5$ and $\int_{- 1}^{5} g (x) d x = - 9$ , find $\int_{- 1}^{5} (5 f (x) + 6) d x$ $61$
Given $\int_{1}^{6} f (x) d x = - 5$ and $\int_{2}^{6} f (x) d x = - 5$ , find $\int_{1}^{2} f (x) d x$ $0$
$f$ is odd and $\int_{0}^{5} f (x) d x = 4$ . Find $\int_{- 5}^{0} f (x) d x$ $- 4$
Given $\int_{- 2}^{3} f (x) d x = - 6$ , $\int_{3}^{5} f (x) d x = 3$ and $\int_{- 2}^{3} g (x) d x = - 2$ , find $\int_{- 2}^{3} [- 2 f (x) - g (x)] d x$ $14$
Given $\int_{1}^{6} f (x) d x = 1$ and $\int_{1}^{6} g (x) d x = - 8$ , find $\int_{6}^{1} (2 f (x) + 3 g (x)) d x$ $22$
Given $\int_{- 1}^{1} f (x) d x = 3$ and $\int_{0}^{1} f (x) d x = - 8$ , find $\int_{0}^{- 1} f (x) d x$ $- 11$
$f$ is even, $g$ is odd, $\int_{0}^{1} f (x) d x = - 1$ and $\int_{0}^{1} g (x) d x = 6$ . Find $\int_{- 1}^{1} (3 f (x) + 4 g (x)) d x$ $- 6$
Given $\int_{- 3}^{3} f (x) d x = - 8$ , $\int_{3}^{7} f (x) d x = 6$ and $\int_{- 3}^{3} g (x) d x = - 1$ , find $\int_{- 3}^{7} f (x) d x$ $- 2$
Given $\int_{1}^{5} - 2 f (x) d x = 18$ , find $\int_{1}^{5} f (x) d x$ $- 9$
Given $\int_{0}^{4} f (x) d x = - 3$ and $\int_{3}^{4} f (x) d x = 7$ , find $\int_{0}^{3} f (x) d x$ $- 10$
Use symmetry to find $\int_{- 5}^{5} (4 x^{3} - 5 x - 6) d x$ $- 60$
Given $\int_{0}^{1} f (x) d x = - 3$ , $\int_{1}^{5} f (x) d x = 1$ and $\int_{0}^{1} g (x) d x = - 9$ , find $\int_{5}^{1} f (x) d x$ $- 1$
Given $\int_{- 2}^{2} f (x) d x = - 8$ and $\int_{- 2}^{2} g (x) d x = - 1$ , find $\int_{- 2}^{2} (- 3 f (x) + 5) d x$ $44$
Given $\int_{- 3}^{3} f (x) d x = - 12$ and $\int_{1}^{3} f (x) d x = 5$ , find $\int_{- 3}^{1} 3 f (x) d x$ $- 51$
$f$ is odd, $g$ is even, $\int_{0}^{4} f (x) d x = - 6$ and $\int_{0}^{4} g (x) d x = 3$ . Find $\int_{- 4}^{0} (- 3 f (x) - 3 g (x)) d x$ $- 27$

How to do these

Given $\int_{1}^{7} f (x) d x = 10$ and $\int_{4}^{7} f (x) d x = 3$ , find $\int_{4}^{1} 2 f (x) d x$

Split the interval: the integral from 1 to 4 is 10 - 3 = 7.
Swapping the limits changes the sign, so the integral from 4 to 1 is -7.
The constant comes out: 2(-7) = -14.

The answer is $- 14$ .

Where students go wrong

Forgetting the length of the interval when a constant is added. The integral of f(x) + 3 from 2 to 6 is the integral of f plus 3 times 4, not plus 3.

Also called rules of definite integrals, definite integral properties, additivity of integrals or integrals of even and odd functions.

What you can put on this worksheet

Properties, from given values: Given $\int_{- 1}^{2} f (x) d x = - 5$ , $\int_{2}^{6} f (x) d x = 3$ and $\int_{- 1}^{2} g (x) d x = 4$ , find $\int_{- 1}^{6} f (x) d x$ → $- 2$
Constant multiples, sums and reversed limits: Given $\int_{1}^{5} - 5 f (x) d x = - 45$ , find $\int_{1}^{5} f (x) d x$ → $9$
Splitting the interval: Given $\int_{- 1}^{5} f (x) d x = - 6$ and $\int_{- 1}^{2} f (x) d x = 3$ , find $\int_{2}^{5} f (x) d x$ → $- 9$
Even and odd functions: $f$ is even and $\int_{0}^{3} f (x) d x = 9$ . Find $\int_{- 3}^{3} f (x) d x$ → $18$

Questions about these worksheets

Yes. Every properties of definite integrals sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the properties of definite integrals questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 4: properties, from given values, constant multiples, sums and reversed limits, splitting the interval and even and odd functions. Tick as many as you want and set how many of each, or let it spread them evenly.

Forgetting the length of the interval when a constant is added. The integral of f(x) + 3 from 2 to 6 is the integral of f plus 3 times 4, not plus 3.

Properties of definite integrals worksheet

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Properties of Definite Integrals

Properties of Definite IntegralsAnswer keyVersion 1

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