Integration is the reversal process that recovers a function from its derivative (or rate of change).
By doing integration it is possible to evaluate displacement over a time period if the velocity function is given.
Initially the reversal approach of integration provides formulas for finding the antiderivatives of common functions. In its advanced approach, integration is viewed as the limit of a sum and used to determine the volumes and areas and is also widely applied in Physics.
Let us look at the definition of integrals, the formulas and rules for integration and its application.

Integral is defined using antiderivatives.

Antiderivative Definition

A function F is an antiderivative of f over an interval, if F'(x) = f(x) for all values of x in the interval.

As adding constants to F(x) does not alter the derivative, F(x) + c is the most general antiderivative of f(x)
and it is commonly known as its integral.

Definition:
Suppose F(x) is an antiderivative of f(x). The indefinite integral of f(x) with respect to x is defined as
$\int f(x)dx$ = F(x) + c,
where c is a constant known as constant of integration.

f(x) is called the integrand, dx defines the variable of integration as x and the process of computing the integral is known as integration.
Example:
We know $\frac{d}{dx}sinx$ = cos x.

Thus sin x is the antiderivative of F(x). The integral of cos x is hence $\int cosx dx$ = sin x + c
where c is the constant of integration.

Integration or antiderivative formulas are got by reversing derivative formulas. The following Table provides the antiderivative formulas to be used in integration process.

$\int x^{n}dx$ = $\frac{x^{n+1}}{n+1}$ + C (n ? -1)

$\int \frac{1}{x}dx$ = ln |x| + C ( x ? 0)

$\int e^{x}dx$ = ex + C

$\int e^{-x}dx$ = -e-x + C

$\int a^{x}dx$ = $\frac{a^{x}}{ln(a)}$ + C

$\int sinx dx$ = -cos x + C

$\int cosx dx$ = sin x + C

$\int sec^{2}xdx$ = tan x + C

$\int csc^{2}xdx$ = -cot x + C

$\int sec(x)tan(x)dx$ = sec x + C

$\int csc(x)cot(x)dx$ = -csc x + C

$\int \frac{1}{\sqrt{1-x^{2}}}dx$ = sin-1 x + C

$\int \frac{1}{1+x^{2}}dx$ = tan-1 x + C

$\int \frac{1}{|x|\sqrt{x^{2}-1}}$ = sec-1 x + C

Integration formulas cannot be derived for many important functions like ln x, tan x cot x etc. We need to use some special techniques to integrate these functions.
The following linearity rules are also applied in integration process in addition to the antiderivative formulas.
Let F(x) and G(x) be the antiderivatives of f(x) and g(x)

1. Constant Multiple Rule:
$\int kf(x)dx$ = k F(x) + C where k is a constant

2. Negative Rule:
$\int -f(x)dx$ = -F(x) + C

3. Sum or Difference Rule:
$\int f(x)\pm g(x)dx$ = F(x) ± G(x) +C

4. Constant Times Variable Rule
$\int f(kx)dx$ = $\frac{F(kx)}{k}$ + C

It can be seen that the negative rule is a special case of constant multiple rule for k = -1.

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The area under a curve is defined to be the limit of the sum of the areas of the partitioned rectangles of equal width Δx.

Area Sum

A definite integral is defined to be limit of such sum and hence the area under a curve in the interval under consideration is
A = $\lim_{n \to \infty }\sum_{i=1}^{n}f(x_{i})\bigtriangleup x$ = $\int_{a}^{b}f(x)dx$
Based on this concept, the definite integrals are used in computing areas , volumes and also used in many Physical formulas to determine the work done, center of gravity etc.
Some of the formulas used applying integration are given below with brief notes.

Area between two curves:

The area between two curves f(x) and g(x) for the interval [a, b] where f(x) ≥ g(x) on [a, b] is given by
A = $\int_{a}^{b}[f(x)-g(x)]dx$

Volume of solid of revolution:

The volume of a solid generated by revolving y = f(x) about x axis on [a, b], using method of disks
V = $\int_{a}^{b}\pi y^{2}dx$

Same way the volume of the solid formed by revolving x = g(y) about y axis for a ≤ y ≤ b, using method
of disks is given by
V = $\int_{a}^{b}\pi x^{2}dy$

The formulas for computing the volume of solid of revolution using the cylindrical shell method are as follows:
V = $\int_{a}^{b}2\pi xf(x)dx$ where 0 ≤ a < b
For computing the volume of solid obtained by rotating
about the y axis, the region under the curve f(x) from
a to b
V = $\int_{a}^{b}2\pi yg(y)dy$ The corresponding formula when the axis of rotation
happens to be the y- axis.

Applications in Physics:

The work done by a force represented by the function f(x) in moving an object from a to b is given by
W = $\int_{a}^{b}f(x)dx$

Velocity function is obtained by integrating the acceleration and the definite integral involving the velocity
function is used in calculating the distance or displacement of a moving object.

Integration formulas are also used to compute the moment, mass and center of mass using the density function.

Integration also finds application in Economics. For example, the cost function can be retrieved from marginal cost using integration.

Solved Examples

Question 1: $\int ($$\frac{2}{x^{2}}$$-x^{2}+5)dx$
Solution:
 
$\int ($$\frac{2}{x^{2}}$$-x^{2}+5)dx$

    =  $\int (2x^{-2}-x^{2}+5)dx$           
        
    =  $\int 2x^{-2}dx$ - $\int x^{2}dx$ + $\int 5dx$                                Sum and difference rules for integration

    = 2$\int x^{-2}dx$ - $\int x^{2}dx$ + $\int 5dx$                                 Constant Multiple rule for integration

    = 2$\frac{x^{-1}}{-1}$ - $\frac{x^{3}}{3}$ + 5x + C                           Integration using power formula for antiderivatives.

    = $\frac{-2}{x}$$\frac{x^{3}}{3}$ + 5x + C   

 

Question 2: $\int $$\frac{t\sqrt{t}+2\sqrt{t}}{\sqrt{t}}$$dt$
Solution:
 
$\int $$\frac{t\sqrt{t}+2\sqrt{t}}{\sqrt{t}}$$dt$

    = $\int ($$\frac{t\sqrt{t}}{\sqrt{t}}$ + $\frac{2\sqrt{t}}{\sqrt{t}})$$dt$            The sum separated with the denominator

    = $\int (t+2)dt$                                                                               Simplified

    = $\frac{t^{2}}{2}$ + 2t + C                                                             Integration using power formula for antiderivatives.
 

Question 3: $\int 7sin(3\theta )d\theta $
Solution:
 
= 7$\int sin(3\theta )d\theta $                                                       Constant multiple rule for integration

    = 7 $(\frac{-cos(3\theta )}{3})$                                                     Integration using the constant times variable rule

    = -$\frac{7cos(3\theta )}{3}$
 

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