Evaluating Definite Integrals Using Properties

Integration and differentiation are the two important process in Calculus. Differentiation is the process of finding the derivative of a function, whereas integration is the reverse process of differentiation. It means that the process of finding the anti-derivative of a function. In integration, the concept behind are functions, limits and integrals. The integrals are generally classified into two different types, namely:

In this article, we are going to discuss the definition of definite integrals, and the process of evaluating the definite integral using different properties.

What is a Definite Integral?

If the upper limit and the lower limit of the independent variable of the given function or integrand is specified, its integration is expressed using definite integrals. A definite integral is denoted as:

\begin{array}{l} F (a) - F (b) = \int_{a}^{b} f (x) d x \end{array}

Here R.H.S. of the equation means integral of f(x) with respect to x.

f(x)is called the integrand.

dx is called the integrating agent.

a is the upper limit of the integral and b is the lower limit of the integral.

Evaluating Definite Integrals – Properties

Let us now discuss important properties of definite integrals and their proofs.

Property 1:

\begin{array}{l} \int_{a}^{b} f (x) d x = \int_{a}^{b} f (t) d t \end{array}

Let us consider x = t. Therefore dx = dt. Substituting these values in the LHS of the above equation we can prove this property.

Property 2:

\begin{array}{l} \int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x \end{array}

According to second fundamental theorem of Calculus, if f(x) is a continuous function defined on the closed interval [a, b] and F(x) denotes the anti-derivative of f(x), then

\begin{array}{l} \int_{a}^{b} f (x) d x = [F (x)]_{a}^{b} = F (b) - F (a) \end{array}

Therefore

\begin{array}{l} \int_{a}^{b} f (x) d x = F (b) - F (a) = - [F (a) - F (b) = - \int_{b}^{a} f (x) d x \end{array}

Property 3:

\begin{array}{l} \int_{a}^{b} d x = \int_{a}^{c} f (x) d x + \int_{c}^{b} f (x) d x \end{array}

From the second theorem of Calculus,

\begin{array}{l} \int_{a}^{b} f (x) d x = [F (x)]_{a}^{b} = F (b) - F (a) \end{array}

Therefore,

\begin{array}{l} \int_{a}^{b} f (x) d x = F (b) - F (a) \dots (1) \end{array}

\begin{array}{l} \int_{a}^{c} f (x) d x = F (c) - F (a) \dots (2) \end{array}

\begin{array}{l} \int_{c}^{b} f (x) d x = F (b) - F (c) \dots (3) \end{array}

Adding equations 2 and 3 we have,

\begin{array}{l} \int_{a}^{c} f (x) d x + \int_{c}^{b} f (x) d x = F (c) - F (a) + F (b) - F (c) = F (b) - F (a) = \int_{a}^{b} f (x) d x \end{array}

Property 4:

\begin{array}{l} \int_{a}^{b} f (x) d x = \int_{a}^{b} f (a + b - x) d x \end{array}

Let us assume that t = a + b – x. Therefore dt = -dx. At x = a, t = b and at x = b, t =a.

Therefore,

\begin{array}{l} \int_{a}^{b} f (x) d x = - \int_{b}^{a} f (a + b - t) d t \end{array}

From the second property of definite integrals:

\begin{array}{l} \int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x \end{array}

Therefore,

\begin{array}{l} \int_{a}^{b} f (x) d x = - \int_{b}^{a} f (a + b - t) d t = \int_{a}^{b} f (a + b - t) d t \end{array}

Following the first property of definite integrals:

\begin{array}{l} \int_{a}^{b} f (x) d x = \int_{a}^{b} f (a + b - x) d x \end{array}

Property 5:

\begin{array}{l} \int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x \end{array}

This property is a special case of fourth property of integrals as discussed above.
Let us assume that t = a – x. Therefore dt = -dx. At x = 0, t = a and at x = a, t =0.

Therefore,

\begin{array}{l} \int_{0}^{a} f (x) d x = \int_{a}^{0} f (a - t) d t \end{array}

From the second property of definite integrals:

\begin{array}{l} \int_{0}^{a} f (x) d x = - \int_{0}^{a} f (x) d x \end{array}

Therefore,

\begin{array}{l} \int_{0}^{a} f (x) d x = - \int_{a}^{0} f (a - t) d t = \int_{0}^{a} f (a - t) d t \end{array}

Following the first property of definite integrals:

\begin{array}{l} \int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x \end{array}

Property 6:

\begin{array}{l} \int_{0}^{2 a} f (x) d x = \int_{0}^{a} f (x) f (x) d x + \int_{0}^{a} f (2 a - x) d x \end{array}

From the second property of definite integrals

\begin{array}{l} \int_{a}^{b} f (x) d x = \int_{a}^{c} f (x) d x + \int_{c}^{b} f (x) d x \end{array}

Therefore,

\begin{array}{l} \int_{0}^{2 a} f (x) d x = \int_{0}^{a} f (x) d x + \int_{a}^{2 a} f (x) d x \dots (4) \end{array}

Let us assume that t = 2a – x. Then dt = – dx. In such a case, when x = 2a, t = 0. Therefore, the second integral can be expressed as:

\begin{array}{l} \int_{a}^{2 a} f (x) d x = \int_{a}^{0} f (2 a - t) d t = \int_{0}^{a} f (2 a - t) d t = \int_{0}^{a} f (2 a - x) d x \end{array}

Equation 4 can, therefore, be rewritten as:

\begin{array}{l} \int_{0}^{2 a} f (x) d x = \int_{0}^{a} f (x) d x + \int_{0}^{a} f (2 a - x) d x \end{array}

Property 7:

\begin{array}{l} \int_{0}^{2 a} \end{array}

f(x)dx= 2

\begin{array}{l} \int_{0}^{a} \end{array}

f(x)dx if f(2a-x)=f(x) and

\begin{array}{l} \int_{0}^{2 a} \end{array}

f(x)dx=0 if f(2a-x)=-f(x)

Using the sixth property of definite integrals, we have

\begin{array}{l} \int_{0}^{2} a \end{array}

f(x)dx=

\begin{array}{l} \int_{0}^{a} \end{array}

f(x)dx+

\begin{array}{l} \int_{0}^{a} \end{array}

f(2a-x)dx——-(5)

If in case f (2a-x)=f(x), therefore equation 5 can be rewritten as:

\begin{array}{l} \int_{0}^{2 a} \end{array}

f(x)dx=

\begin{array}{l} \int_{0}^{a} \end{array}

f(x)dx+

\begin{array}{l} \int_{0}^{a} \end{array}

f(x)dx

If f(2a – x) = -f(x), then equation 5 can be rewritten as:

\begin{array}{l} \int_{0}^{2 a} \end{array}

f(x)dx=

\begin{array}{l} \int_{0}^{a} \end{array}

f(x)dx -

\begin{array}{l} \int_{0}^{a} \end{array}

f(x)dx

Property 8:

\begin{array}{l} \int_{- a}^{a} \end{array}

f(x)dx=2

\begin{array}{l} \int_{0}^{a} \end{array}

f(x)dx if f(-x)=f(x) and

\begin{array}{l} \int_{- a}^{a} \end{array}

f(x)dx=0 if f(-x)= -f(x)

Using third property of definite integrals we have

\begin{array}{l} \int_{a}^{b} \end{array}

f(x)dx=

\begin{array}{l} \int_{a}^{c} \end{array}

f(x)dx +

\begin{array}{l} \int_{c}^{b} \end{array}

f(x)dx

Therefore,

\begin{array}{l} \int_{- a}^{a} \end{array}

f(x)dx=

\begin{array}{l} \int_{- a}^{0} \end{array}

f(x)dx+

\begin{array}{l} \int_{0}^{a} \end{array}

f(x)dx——–(6)

Let us assume that t = -x. Then dt = -dx. Thus when x = -a, t = a and x =0, t = 0. Then the first integral in right hand side can be written as:

\begin{array}{l} \int_{- a}^{0} \end{array}

f(x)dx= -

\begin{array}{l} \int_{a}^{0} \end{array}

f(-t)dt

Therefore equation 6 can be written as;

\begin{array}{l} \int_{- a}^{a} \end{array}

f(x)dx= -

\begin{array}{l} \int_{a}^{0} \end{array}

f(-t)dt +

\begin{array}{l} \int_{0}^{a} \end{array}

f(x)dx

\begin{array}{l} \int_{- a}^{a} \end{array}

f(x)dx=

\begin{array}{l} \int_{0}^{a} \end{array}

f(-x)dx +

\begin{array}{l} \int_{0}^{a} \end{array}

f(x)dx———(7)

In case if f is an even function, i.e., f(-x) = f(x), then equation 7 can be rewritten as:

\begin{array}{l} \int_{- a}^{a} \end{array}

f(x)dx=

\begin{array}{l} \int_{0}^{a} \end{array}

f(x)dx+

\begin{array}{l} \int_{0}^{a} \end{array}

f(x)dx= 2

\begin{array}{l} \int_{0}^{a} \end{array}

f(x)dx

In case if f is an odd function, i.e., f(-x) = – f(x), then equation 7 can be rewritten as:

\begin{array}{l} \int_{- a}^{a} \end{array}

f(x)dx=

\begin{array}{l} \int_{0}^{a} \end{array}

f(x)dx-

\begin{array}{l} \int_{0}^{a} \end{array}

f(x)dx= 0

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MATHS Related Links
Roman Letters 1 To 100	Supplementary Angles
Properties Of Trapezium	Area Of Cuboid Formula
Number System In Maths	Rational Numbers Examples
Divisibility Rule Of 7	Area Of The Rectangle
Surface Area And Volume Class 10	Real Numbers

MATHS Related Links

Roman Letters 1 To 100

Supplementary Angles

Properties Of Trapezium

Area Of Cuboid Formula

Number System In Maths

Rational Numbers Examples

Divisibility Rule Of 7

Area Of The Rectangle

Surface Area And Volume Class 10

Real Numbers

Evaluating Definite Integrals

What is a Definite Integral?

Evaluating Definite Integrals – Properties

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