1. Why These Properties Are Useful
The properties of definite integrals help simplify expressions, break integrals into manageable pieces, and identify quick shortcuts. In personal-notes style: these are the tools that save time and avoid unnecessary calculations.
2. Reversal of Limits
Switching the upper and lower limits changes the sign of the integral.
2.1. Formula
\int_a^b f(x) dx = -\int_b^a f(x) dx
2.2. Quick Note
Useful when limits appear in the wrong order or when matching integrals for simplification.
3. Zero-Length Interval
If both limits are the same, the integral is zero because the interval has no width.
3.1. Formula
\int_a^a f(x) dx = 0
4. Additivity Over Intervals
A definite integral can be broken into pieces at any point between the limits.
4.1. Formula
\int_a^c f(x) dx = \int_a^b f(x) dx + \int_b^c f(x) dx
4.2. Use Case
Helpful when integrating piecewise or symmetric functions.
5. Linearity of Definite Integrals
The integral distributes over addition and allows constants to be pulled outside.
5.1. Formulas
\int_a^b [f(x) + g(x)] dx = \int_a^b f(x) dx + \int_a^b g(x) dx
\int_a^b c f(x) dx = c \int_a^b f(x) dx
6. Symmetry Property — Even Functions
If the function is even (f(−x) = f(x)), the integral over symmetric limits can be doubled.
6.1. Formula
\int_{-a}^{a} f(x) dx = 2 \int_0^a f(x) dx
6.2. Example
\int_{-2}^{2} x^2 dx = 2 \int_0^{2} x^2 dx
7. Symmetry Property — Odd Functions
If the function is odd (f(−x) = −f(x)), the integral over symmetric limits is zero.
7.1. Formula
\int_{-a}^{a} f(x) dx = 0
7.2. Example
\int_{-3}^{3} x^3 dx = 0
8. Comparison Property
If one function is always larger than another on the interval, its integral is also larger.
8.1. Statement
\text{If } f(x) \le g(x) \text{ on } [a, b], \text{ then } \int_a^b f(x) dx \le \int_a^b g(x) dx
9. Examples Using Properties
These examples show how properties help simplify work before integrating.
9.1. Example 1 — Reversing Limits
\int_5^2 x dx = -\int_2^5 x dx
9.2. Example 2 — Symmetry
\int_{-\pi}^{\pi} \sin x dx = 0 \quad \text{(odd function)}
9.3. Example 3 — Even Function Trick
\int_{-4}^{4} (x^4 + 1) dx = 2 \int_{0}^{4} (x^4 + 1) dx
10. Why These Properties Matter
These properties help break tough integrals into simpler ones, handle symmetry problems quickly, and reduce errors. They are especially helpful in definite integrals without needing full computation.