Properties of Definite Integral

Identical Integration Limits

If the upper and lower limits of a definite integral are the same, the result is zero.

\int_{a}^{a} f(x) \, dx = 0

A definite integral accumulates a quantity across an oriented interval. From $a$ to the same point $a$ , the interval has zero width, so the accumulation is zero regardless of the value of the integrand.

Reversing the Integration Limits

When we swap the lower and upper limits of an integral, the result is the negative of the original integral's value.

\int_{a}^{b} f(x) \, dx = - \int_{b}^{a} f(x) \, dx

The lower and upper bounds determine the orientation of the integration interval. Swapping the bounds reverses that orientation and changes the sign of every contribution to the accumulation. Therefore, the integral becomes the negative of its original value.

Constant Multiple Rule

Just as with indefinite integrals, a constant can be factored out of the integral to simplify the calculation.

\int_{a}^{b} kf(x) \, dx = k \int_{a}^{b} f(x) \, dx

Multiplying every value of a function by a constant $k$ multiplies its accumulated integral by the same factor. We may therefore evaluate $\int f(x)dx$ first and then multiply by $k$ .

Sum and Difference Rule

The integral of a sum or difference of two functions is equal to the sum or difference of their individual integrals.

\int_{a}^{b} [f(x) + g(x)] \, dx = \int_{a}^{b} f(x) \, dx + \int_{a}^{b} g(x) \, dx

By linearity, a complicated integrand can be split into simpler pieces. Integrate $f(x)$ and $g(x)$ separately, then combine their signed accumulations with the same operation.

Interval Addition Property

An integration interval can be split at an intermediate point. The signed accumulation over the whole interval equals the sum over the two parts.

\int_{a}^{c} f(x) \, dx = \int_{a}^{b} f(x) \, dx + \int_{b}^{c} f(x) \, dx

This identity holds for any order of $a$ , $b$ , and $c$ , provided the function is integrable on the required interval. In words, the accumulation from $a$ to $c$ equals the accumulation from $a$ to $b$ plus the accumulation from $b$ to $c$ . It is especially useful for piecewise functions because each formula can be integrated on the interval where it applies.