The properties of indefinite integrals apply to definite integrals as well. Definite integrals also have properties that relate to the limits of integration. These properties, along with the rules of integration help us manipulate expressions to evaluate definite integrals.

Property 1 :

If the limits of integration are the same, the integral is just a line and contains no area.

Property 2 :

If the limits are reversed, then place a negative sign in front of the integral.

Property 3 :

The integral of a sum is the sum of the integrals.

Property 4 :

The integral of a difference is the difference of the integrals.

Property 5 :

The integral of the product of a constant 'c' and a function f(x) is equal to the constant multiplied by the integral of the function.

Property 6 :

Although this formula normally applies when c is  between a and b (a < x < b), the formula holds for all values of a, b, and c, provided f(x) is integrable on the largest interval.

Property 7 :

For the same function and same limits, if the variable is changed, the result of a definite integral will not change.

Property 8 :

In a function f(x),

f(-x) = -f(x) ----> f(x) is an odd function

f(-x) = f(x) ----> f(x) is an even function

If f(x) is an odd function,

If f(x) is an even function,

Property 9 :

If f(a + x) = f(x), f(x) is a period function with the period is a.

Property 10 :

Solved Problems

Problem 1 :

Evaluate :

Solution :

Using the property 10,

Adding (1) and (2),

Problem 2 :

Evaluate :

Solution :

f(x) is an even function.

Let k = a5 and u = x5.

ᵈᵘ⁄d = 5x4

du = x4dx

When x = 0,

u = 0

When x = 0,

u = 25 = 32

Formula :

Substitute k = a5.

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