## Properties of Definite Integrals

### Properties of Definite Integrals are certain key properties that help us manipulate and evaluate definite integrals

Properties of Definite Integrals are certain key properties that help us manipulate and evaluate definite integrals. Here are some important properties:

1. Linearity: The definite integral is a linear operator, which means it obeys the properties of linearity. For any functions f(x) and g(x), and real numbers a and b, we have:

∫[a,b] (af(x) + bg(x)) dx = a∫[a,b] f(x) dx + b∫[a,b] g(x) dx

2. Change in Limits: Changing the limits of integration can be done by adding or subtracting a constant.

∫[a,b] f(x) dx = -∫[b,a] f(x) dx

∫[a,b] f(x) dx = ∫[c,d] f(x) dx + ∫[d,b] f(x) dx

3. Splitting the Interval: If we have a definite integral over an interval [a, b] that contains a sub-interval [c, d], we can split the integral into two separate integrals.

∫[a,b] f(x) dx = ∫[a,c] f(x) dx + ∫[c,d] f(x) dx + ∫[d,b] f(x) dx

4. Reversing the Limits: Reversing the limits of integration changes the sign of the integral.

∫[a,b] f(x) dx = -∫[b,a] f(x) dx

5. Integration by Substitution: If we have a function f(g(x)) and its derivative is f'(g(x)), then we can use substitution to evaluate the integral.

∫[a,b] f(g(x)) * g'(x) dx = ∫[c,d] f(u) du

6. Symmetry: If f(x) is an even function, meaning f(-x) = f(x), then the definite integral over a symmetric interval is zero.

∫[-a,a] f(x) dx = 0

7. Additivity: If we have two intervals [a, b] and [b, c], then the sum of the integrals over these intervals is equal to the integral over the combined interval [a, c].

∫[a,b] f(x) dx + ∫[b,c] f(x) dx = ∫[a,c] f(x) dx

These properties are useful for simplifying complicated integrals, evaluating definite integrals, and solving problems in various areas of mathematics and science.

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