Numerical Integrals

The process of finding an integral; it may be definite integral or an indefinite integral is referred as integration. Numerical integration is mainly used for finding the numerical value of a definite integral. Numerical integration is also used to finding the numerical solution of differential equations. Numerical integration is also referred to as numerical quadrature.

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Numerical integrals:- types of integrals

1. Indefinite integrations:

An indefinite integration is the family of functions that have a given function as a common derivative. The indefinite integral of f(x) is written ∫ f(x) dx.

2. Definite integrations:

If F(x) is the integral of function f(x) over the interval [a, b] ,i.e., ∫ f(x) dx = F(x) then the definite integral of function  f(x) over the interval [a, b] is denoted by int_a^bf(x)dx and is defined as int_a^bf(x)dx  = F(b) – F(a).

Where 'a' is called the lower limit and b is called the upper limit of integration and the interval [a, b] is called of integration.

Methods of integration:

It is not possible to integrate each integral with help of the following methods but a large number of varieties of the problems can be solved by these methods so, we have the following methods of integration:

1. Integration by substitution

2. Integration by parts.

3. Partial fractions

Example Problems on indefinite and definite integration:

Numerical Integrals Problem:

Example 1:

find ∫ 1 / sin2x cos2x dx.

Solution:

We have ∫ 1/ sin2x cos2x dx

= ∫sin2x+cos2x / sin2x+cos2x dx

=∫ (1/cos2x+1/ sin2x )dx

=∫sec2 x d x+∫cosec 2 x dx

=tan x -cot x+C.

Example 2:

∫ sin x sin (cos x) dx.

Solution:

Let cos x = t

dt = -sin x dx

Therefore we have

∫sin x sin (cos x) dx = - ∫sint dt = cost + C=cos (cos x) + C.

where

t=cos x

Example 3:  int_0^1dx/(1+x2)

Solution:

=[tan-1 x]10

=tan-1 1 - tan-1 0

= /4 -0 = / 4

π/2

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Example 4: Find the value int sin⁷xdx

-π/2

Solution:

Let f (x) = dx.

Then f(-x) = -sin7 x=-f (x)

so, f (x) is odd function.

π/2

π/2

Therefore, int f(x)dx=0 => int sin7x dx = 0.

-π/2                  -π/2