Properties of Integrals

Computing integrals gives us either numerical values or a new function whose derivative is the original function. Integrals can either be definite or indefinite. The properties of these integrals help simplify the process of integration.
Properties of definite integrals:

Definite integrals are written in the form ∫abf(x)dx. Their properties include:

1. Linearity Property:

∫ab[f(x)+g(x)]dx = ∫abf(x)d(x) + ∫ab g(x)dx

∫abc · f(x)dx=c· ∫abf(x)dx

According to this property, we can split the integral of a sum or difference into separate integrals. The constants can be taken out of the integrals.

2. Reversal of Limits:

abf(x)dx=-baf(c)dx
Interchanging the limits of the integration results in changing the sign of the result.

3. Zero Interval Property

aaf(x)dx=0
When the upper and lower limits are the same, the area is zero because there is no interval.

4. Additivity Over Intervals

∫acf(x)dx=∫abf(x)dx+bcf(x)dx    (a <b<c)
This property shows that an integral can be split across a point inside the interval.

5. Even Function Property

∫-aaf(x()dx=2∫0af(x)dx   if f(x) =f(-x)

If the function is symmetric about the y-axis, this means the area is equal on both sides.

6. Odd Function Property

∫-aaf(x)dx=0  if  f(x)= -f(-x)

The property states that positive and negative parts cancel out for symmetric limits.

7. Non-negativity Property

If f(x) ≥ 0 on [a,b] then,
 
∫abf (x) dx ≥ 0

The area under the curve cannot be negative if the function is always above the x-axis.

An indefinite integral gives the antiderivative and a constant C. It also represents a family of functions and is written as f(x)dx. Their properties are listed below:
Linearity Property:
Like definite integrals, we can split integrals or factor constants out in indefinite integrals as well.
f(x)g(x)dx=f(x)dxg(x)dx
cf(x)dx=cf(x)dx

Power Rule:
This is a basic rule that must be followed while finding the antiderivatives of powers of x.
xndx=xn+1n+1+C   (n -1)

Constant Rule:
c dx=cx+C  (n-1)
According to this property, the integral of a constant is the product of the constant and the variable.

Zero Function Rule:
0dx=C
If there is nothing to integrate, then the constant is the answer of the integration.

Reversal of Differentiation
ddxf(x)dx=f(x)
This property establishes that integration is the exact reverse of differentiation.

General Antiderivative
If F(x) is an antiderivative of f(x), then,
f(x)dx=F(x)+C
The property suggests that there are an infinite number of antiderivatives that vary only by a constant C

Properties of integrals can often seem intimidating to beginners. Here are some useful tips and tricks to help you gain a strong understanding of them.

• Use linearity to simplify complex expressions; doing so makes them easier to solve.

• Factoring out constants from the integral simplifies calculations and reduces mistakes.

• Use interval splitting, breaking the Integral over multiple parts, helps when the function changes form in the interval.

• For simple functions like lines, rectangles, or triangles, use geometry instead of integration. This will save you time.

• Do not forget to add the constant in indefinite integrals. It is important for general solutions and value problems.

• Integration: The process of computing the integral of a function is known as integration. It can also be defined as the reverse of differentiation.

• Differentiation: The process of finding the rate at which a function changes for its variable is known as differentiation.

• Derivative: A derivative is the result of differentiation. It represents the rate of change of a function.

• Antiderivative: An antiderivative is a function that, upon computing its derivative, gives the original function.

• Continuity: In integration, continuity means that a function has no breaks over the interval being integrated.