Definite Integral Formulas

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Concept of Definite Integrals

The definite integral is closely linked to the antiderivative and indefinite integral of a given function. The introduction of the concept of a  definite integral of a given function initiates with a function f(x) which is continuous on a closed interval (a,b). The interval which is given is divided into “n” subinterval is that, although not mandatory can be considered of equal lengths(Δx). An arbitrary domain value xi is selected in each subinterval and its corresponding subinterval length is calculated and these ‘n’ products are added to calculate their sums. The sum is known as Riemann sum and may be either positive, negative or zero relies upon the behavior of the function on a closed interval.

For example, if (fx) is greater than 0 on [a,b] then the Riemann sum will be the positive real number and if (fx) is lesser than 0 on [a,b], then the Riemann sum will be the negative real number. The Riemann sum of the function f( x) on [ a, b] is represented as as

Sn = f(x1) Δx + f(x2)Δx+ f(x3) Δx+…. + f(xn) Δx

or Sn = \[\sum_{i=1}^{n}\] (fxi ) Δx


What is Integration in Maths?

In Maths, integration is a process of summing up parts to determine the whole. It is a reverse process of differentiation where we can split the function into parts. Integration is used to determine the summation under a very large scale. Calculating a total of small numbers is an easy task and can be done even manually, but calculating a total of large numbers where the limit could reach infinity is a complex task. In such a case, an integration method is used. Integration and Differentiation both are important concepts of calculus. There are two different types of integration namely:

  • Definite Integral

  • Indefinite integral 

This article delivers information about the concepts of definite integrals, definite integrals equations, properties of definite integrals, definite integration by parts formula, reduction formulas in definite integration etc.


Definite Integral Equation

An integral including both upper and lower limits is considered as a definite integral. A Riemann integral is considered as a definite integral where x is confined to fall on the real line.

\[\int_{a}^{\infty}\] f(x) dx = \[\lim_{b\rightarrow\infty}\] [\[\int_{a}^{b}\] f(x) dx]

[\[\int_{a}^{b}\] f(x)  dx = F(b) - F(a)

In the above definite integral equation a,∞, and b are  determined as the lower and upper limits, F(a) is considered as the lower limit value of the integral and F(b) is considered as the upper limit value of the integral

Definite Integration by Parts Formula

Definite integration by parts formula is generally used to integrate the product of two functions. The definite integration by parts formula is given as :

\[\int\] p q dx = p \[\int\] q dx - \[\int\] p’ (\[\int\] q dx ) dx

In the above Definite integration by parts formula.

p represents the function p(x)

q represents the function q (x)

p’ is derivative of the function p(x).


Reduction Formula in Definite Integration

Some of the reduction formulas in definite integration are:

  • Reduction formula for sin - Sinn x dx = -1/n cos x sinn-1 x + n-1/n \[\int\] sinn-2 x dx

  • Reduction formula for cos = Cosn x dx = -1/n sin x cosn-1 x + n-1/n \[\int\] cosn-2 x dx

  • Reduction formula for natural logarithm - \[\int\] (In x)n dx = x(In x)n – n \[\int\] (In x)n-1 dx


Properties of Definite Integrals

Some of the properties of definite integrals are given below:

  1. \[\int_{a}^{b}\] f (x) dx = - \[\int_{b}^{a}\] f (x) dx 

  2. \[\int_{a}^{a}\] f (x) dx = 0

  3. \[\int_{a}^{b}\]k f (x) dx = k\[\int_{a}^{b}\] f (x) dx 

  4. \[\int_{a}^{b}\] f (x) ± g(x) dx = \[\int_{a}^{b}\] f (x) dx ± \[\int_{a}^{b}\] g(x) dx

  5. \[\int_{a}^{b}\] f (x) dx = \[\int_{a}^{c}\] f (x) dx + \[\int_{c}^{b}\] f (x) dx

  6. \[\int_{a}^{b}\] f (x) dx = \[\int_{a}^{b}\]f (t) dt


Definite Integral Solved Examples of Definite Integral Formulas

Some of the solved examples of definite integrals are given below:


1. Given that, \[\int_{0}^{3}\] x² dx = 8 , solve \[\int_{0}^{3}\] 4x² dx


Solution: 

\[\int_{0}^{3}\] 4x² dx =  4 \[\int_{0}^{3}\]x² dx

= ( 4) × 8

= 32


2. Solve \[\int_{2}^{6}\] 3 dx


Solution:

\[\int_{2}^{6}\]  3 dx = 3(6-2)

= 12


3. Evaluate \[\int_{1}^{2}\] xdx/(x² + 2)³


Solution:

 By using the substitution method, we get 

u = x² + 2

du = 2x dx

½ du = x dx

The limit of integration can be transformed from x values to their corresponding  values. When x= 1,u = 3 and when x =2 , u = 6, find 

\[\int_{1}^{2}\] xdx/(x² + 2)³ = ½ \[\int_{3}^{6}\] du/u³ 

= ½ \[\int_{3}^{6}\] u-3 du

= ½ [ -½ u-2 ] \[\int_{3}^{6}\]

= -¼ [(6)-2 - (3)-2]

= -¼ (1/36 - 1/9)

= 1/48

It is important to note that the substitution method is used to calculate definite integrals and it is not necessary to return back to the original variable if the limit of integration is transformed to the new variable values.’


Quiz Time

1. \[\int_{0}^{10y}\] |sin y| dy is

  1. 20

  2. 8

  3. 10

  4. 18


2. \[\int_{-1.2}^{1/2}\] cos y In ( 1+ y/1-y)dy is Equal to

  1. 1

  2. 2

  3. 3

  4. None of these

FAQ (Frequently Asked Questions)

1. What are the Uses of Definite Integrals?

Definite integrals are used in different fields. It is applied in economics, finance, engineering, and physics. For example, marginal cost yields cost, income rates obtain total income, velocity accrues to distance, and density yields volume. The definite integrals is also used to bring forth operations on functions such as calculating arc, length, volume, surface areas and many more. is also used to determine the area under a curve and area between two curves. Some examples of definite integral where it is generally used are line integral, surface integral, and contour integrals.

2. Explain the Meaning of Definite Integral and Indefinite Integral.

The definite integral f(k) is a number that denotes area under the curve f(k) from k = a and k = b.

A definite integral retains both lower limit and the upper limit on the integrals and it is known as a definite integral because, at the completion of the problem, we get a number which is a definite answer. The definite integral is expressed as

abf(x) dx


Indefinite Integral 

The definite integral f(x) is a function that obtains the answer of the question “ What function when differentiated gives f(x).

An indefinite integral has no lower limit and the upper limit on the integrals and obtains the answer that has variable x in it and also retains constant value (usually represented by C) in it.

Indefinite integral generally provides a general solution to the differential equation. It is basically a general form of integration and can be expressed as the antiderivative of the considered function.  The indefinite integral is expressed as

∫f(x) dx