Integrals

Integration is widely used in engineering and science to find the area of curves, volume of solids, the lateral surface area of solids and other astrophysical applications.

Definition of Integration

The inverse process of differentiation is called integration

Let F(x) be a differentiable function of ‘x’ such that d/dx[F(x)] = f(x) then the function F (x) is called a primitive or anti derivative or indefinite integral or simply integral of f(x) with respect to ‘x’ and is written symbolically as $\int$  f ( x) dx = F (x) + C or $\frac{d}{dx}$ [F(x) + C]= f(x)

Where f(x) is called the integrand and ‘C’ and is called the constant of Integration, may be real or imaginary.


Table of integrals [Integration Table]


Function Integral of the function 
$\int$ xn dx  $\frac{(xn+1)}{( n + 1 )}$ + C
$\int \frac{1}{2}$ dx ln|x| + C
$\int e^x$ dx
ex + C
$\int a^x$ dx
$\frac{ax}{In(a)}$+ C
$\int$ 1 dx x + C
$\int$ cos (x) dx
sin ( x ) + C
$\int$ Tan (x) dx - log | cos ( x ) | + C
$\int$ Cot (x) dx log | sin ( x ) | + C
$\int$ Sec (x) dx log | sec ( x ) + tan ( x ) | + C
$\int$ Csc (x) dx log | css ( x ) - cot ( x ) | + C
$\int$ Sin (x) dx -  cos ( x ) + C
$\int\frac{1}{\sqrt{a^2-x_2}}$ dx sin-1$\frac{1}{a}$ + C
$\int\frac{-1}{\sqrt{a^2-x_2}}$ dx cos-1($\frac{x}{a}$) + C
$\int\frac{1}{a^2+x_2}$ dx tan-1($\frac{x}{a}$) + C
$\int\frac{-1}{a^2+x_2}$ dx cot-1($\frac{x}{a}$) + C
$\int\frac{-1}{|x|\sqrt{a^2-x_2}}$ dx
(1/a) * csc-1($\frac{x}{a}$) + C
$\int\frac{1}{|x|\sqrt{a^2-x_2}}$ dx (1/a) * sec-1($\frac{x}{a}$) + C
$\int$ sinh (x) dx cosh ( x ) +
$\int$ cosh ( x ) dx sinh ( x ) + C
$\int$ tanh ( x ) dx ln cosh ( x ) + C
$\int$ coth ( x ) dx ln |sinh ( x )| + C
$\int$ Sech ( x ) dx tan-1 |sinh ( x )| + C
$\int$ Csch2 ( x ) dx -coth ( x ) + C


Common Integrals

Given below are some common integrals:
  •    Integration of 1 is equal to ‘ x ’ + C
  •    Integration of xn is equal to ‘ $\frac{(xn + 1)}{(n + 1)}$ + C ‘ where n is not equal to 1 
  •    Integration of $\frac{1}{x}$ is equal to ‘ ln |x| ’ + C
  •    Integration of ($\frac{1}{ax + b}$) is equal to ‘ $\frac{1}{a}$ * ln (ax + b) ’ + C

Trigonometric Integrals

Given below are some trigonometric integrals:
  •    Integration of sin ( x ) is equal to ‘ -  cos ( x ) ’ + C
  •    Integration of cos ( x ) is equal to ‘ sin ( x ) ’ + C
  •    Integration of tan ( x ) is equal to ‘- log | cos ( x ) | ’ + C
  •    Integration of cot ( x ) is equal to ‘ log | sin ( x ) | ’ + C
  •    Integration of sec ( x ) is equal to ‘ log | sec ( x ) + tan ( x ) | ’ + C
  •    Integration of csc ( x ) is equal to ‘ log | css ( x ) - cot ( x ) |’ + C
  •    Integration of sec2 ( x ) is equal to ‘ tan ( x ) ’ + C
  •    Integration of csc2 ( x ) is equal to ‘ - cot ( x ) ’ + C
  •    Integration of sec ( x ) tan( x )  is equal to ‘ sec ( x ) ’ + C
  •    Integration of csc ( x ) cot ( x ) is equal to ‘ - csc ( x ) ’ + C

Integration by Parts

Let u(x) and v(x) be two functions of ‘x’, then we know that from differential calculus
D[u(x) * v(x) ] = u(x) * v’(x) + v(x) * u’(x)

Integrating both sides with respect to ‘x’, we get

 uv = $\int u \frac{dv}{dx}$dx+$\int v \frac{du}{dx}$ dx

= $\int u \frac{dv}{dx}$ dx = uv - $\int u \frac{du}{dx}$
  - - - - - - - - - - - - - - - - - - - - - - - (i)

Now put u = f( x ) and v =  g ( x ) dx, so that $\frac{dv}{dx}$ = g( x )

Substituting these values in equation (i), we get

     f( x). g( x ) dx = f( x ) * {$\int$  g (x) dx} - $\int$ [f’ ( x ) * {$\int$ g ( x ) dx}] dx

The above formula can be put in words that is

The integral of the product of two functions = (first function) x (integral of the second function) – Integral of the derivative of the first function x Integral of the second function)

Definite Integrals

The Newton – Leibnitz formula ( or ) The fundamental Theorem of Integral calculus

If y(x) is one of the primitives or anti derivatives of a function f (x) defined on [a, b], then the definite integral of f(x) over [a, b] is given by y (b) – y (a) and is denoted by $\int_{a}^{b} f(x) dx$  read as integral of f(x) dx from ‘a’ to ‘b’.

The numbers ‘a’ and ‘b’ are called the limits of integration ‘a’ is called the lower limit and ‘b’ is called the upper limit. The interval [a, b] is called the interval of integration.

In definite integrals constant of integration does not exist.

Improper Integrals

Definition of Improper Integral
If f ( x ) is continuous on [a,∞ ] then $\int_{0}^{\infty } f(x) dx$ is called an improper integral and is defined as

$\int_{0}^{\infty } f(x) dx$ = $\lim_{b \to \infty } \int_{a}^{b}f(x) dx$  - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - (i)

This above equation is read as integral f(x) dx from ‘0’ to infinity is equal to limit b tends to infinity integral f(x) dx from a to b

If there exists a finite limit on the right hand side of (i) then we say that the improper integral is convergent, otherwise it is divergent

Similarly, we define $\int_{0}^{-\infty } f(x) dx$ = $\lim_{b \to \infty } \int_{a}^{b}f(x) dx$ and $\int_{\infty}^{-\infty } f(x) dx$ = $\int_{\infty}^{-\infty } f(x) dx+\int_{a}^{-\infty } f(x) dx$

Geometrically, for f ( x ) > 0 the improper $\int_{a}^{-\infty } f(x) dx$ integral  gives area of the figure bounded by the curve y = f ( x ), the x – axis and the straight line x = a.

Special Integrals

(i)    Sometimes we split the integrand into the sum of two parts such that the integration of one of them by parts cancels the other
$\int$  ex {f ( x ) + f’( x )} dx = ex f ( x ) + c

(ii)        Following are the trigonometric substitutions that are useful in evaluating integrals

S. No
Expression
Substitution
1
a2 + x2
x = a tanθ  or x = a cotθ  
2
a2 - x2
x = a sinθ  or x = a cosθ
3
x2 - a2
x = a secθ  or x = a cscθ
4
$\sqrt{[\frac{(a – x)}{(a + x)}]}$or$\sqrt{[\frac{(a + x)}{(a - x)}]}$
x = a cos2θ
5
$\sqrt{[\frac{(x-p)}{(q- x)}]}$
x = p cos2θ  + q sin2θ


Here are some special integrals formula:
   
  •    Integration of 1 divided by square root of ( a2 – x2 ) is equal to ‘ sin-1($\frac{x}{a}$) ’ + C
  •    Integration of - 1 divided by square root of ( a2 – x2 ) is equal to ‘ cos-1($\frac{x}{a}$) ’ + C
  •    Integration of 1 divided by ( a2 + x2 ) is equal to ‘ 1/a * tan-1($\frac{x}{a}$) ’ + C
  •     Integration of - 1 divided by ( a2 + x2 ) is equal to ‘ cot-1($\frac{x}{a}$) ’ + C
  •     Integration of 1 divided by x times square root of ( x2 – a2 ) is equal to ‘ ($\frac{1}{a}$) *sec-1($\frac{x}{a}$) ’ + C
  •     Integration of - 1 divided by x times square root of ( x2 – a2 ) is equal to ‘ (1/a) *csc-1($\frac{x}{a}$) ’ + C
  •     Integration of 1 divided by ( a2 - x2 ) is equal to ‘ $\frac{1}{2}$a * |$\frac{log|(x - a)}{(x + a)}$|’ + C
  •     Integration of 1 divided by ( x2 - a2 ) is equal to ‘ $\frac{1}{2a}$ * |$\frac{log|(a+x)}{(a-x)}$| ’ + C
  •     Integration of 1 divided by $\sqrt{a^2+x^2}$ is equal to ‘ log | x + $\sqrt{ (a^2 + x^2)}$ | ’ + C
  •     Integration of 1 divided by $\sqrt{x^2-a^2}$ is equal to ‘ log | x + $\sqrt (x^2 - a^2)$ | ’ + C
  •     Integration of $\sqrt{a^2-x^2}$ is equal to ‘ $\frac{1}{2}x$ $\sqrt(a^2 - x^2)+\frac{1}{2}a2 sin-1\frac{x}{a}$ ’ + C
  •     Integration of $\sqrt({a^2+x^2)}$ is equal to ‘ $\frac{1}{2}x\sqrt(a^2+x^2 )+\frac{1}{2}a2 log | x + \sqrt(a^2+ x^2)$ |’ + C
  •     Integration of $\sqrt{x^2-a^2}$ is equal to ‘ $\frac{1}{2}x \sqrt(x^2-a^2 ) - \frac{1}{2}a2 log | x + \sqrt(x^2-a^2)$ |’ + C

Integrals of Hyperbolic Trigonometric functions


  •    Integration of sinh ( x ) is equal to ‘ cosh ( x ) ’ + C
  •     Integration of cosh (x) is equal to ‘ sinh (  x ) ’ + C
  •     Integration of tanh ( x ) is equal to ‘ ln cosh ( x ) ’ + C
  •     Integration of coth ( x ) is equal to ‘ ln |sinh ( x )| ’ + C
  •     Integration of sech ( x ) is equal to ‘ tan-1 |sinh ( x )| ’ + C
  •     Integration of sech( x ).tanh( x ) is equal to ‘ - sech ( x ) ’ + C
  •     Integration of cssh( x ).coth( x ) is equal to ‘ - csch ( x ) ’ + C

Properties of Integral

Here are properties of integrals:

1. $\int_{a}^{b}(f(x)+g(x))+dx$   =$\int_{a}^{b} f(x) dx$ = $\int_{a}^{b} g(x) dx$

2. $\int_{a}^{b} k. f(x) dx$ = $k\int_{a}^{b} f(x) dx$

3. $\int_{a}^{b} f(x) dx$ = $\int_{a}^{b} f(t) dt$

4. $\int_{a}^{b} f(x) dx$  = $\int_{a}^{b} f(x) dx$

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

6. $\int_{a}^{b} f(x) dx$  = $\int_{a}^{b} f(a+b-x) dx$

7. $\int_{0}^{a} f(x) dx$  = $\int_{0}^{a} f(a-x) dx$

8. $\int_{a}^{b} f(x) dx$  = $\int_{a-c}^{b-c} f(x+c) dx$

 $\int_{a}^{b} f(x) dx$  = $\int_{a+c}^{b+c} f(x-c) dx$

9.  $\int_{a}^{b} f(x) dx$ = $k\int_{\frac{a}{k}}^{\frac{b}k{}} f(kx)dx$

 $\int_{a}^{b} f(x) dx$ = $\frac{1}{k} \int_{a}^{b} f(\frac{x}{k}) dx$

10. $\int_{a}^{b} f(x) dx$ = $\int_{-a}^{-b} f(-x) dx$   

11.  $\int_{0}^{2a} f(x) dx$ = $\int_{0}^{a} {f(a-x)+f(a+x)} dx$

12.  $\int_{a}^{b} f(x) dx$ = $(b-a)\int_{0}^{1} {f(b-a)x+a} dx$

Types of Integrals

The different types of integrals (list of integrals) are:1.    Double Integrals
2.    Triple Integrals
3.    Line Integrals
4.    Surface Integrals
Double integrals   

The concept of the definite integral could be extended to functions that have more than one variable.

Let us take, a function of two variables z = f (x, y). The double integral of this function f (x, y) is written as
  $\int\int_{p} f(x,y) da $

Where ‘ P ‘ is the region of integration in the xy - plane.

If $\int_{a}^{b}f(x) dx$  is the definite integral of a function f(x) in one variable such that f (x) ≥ 0 is the area under the curve f (x) from x = a to x = b, then the double integral will give the volume under the surface of z = f (x, y) and the xy  - plane in the region of integration ‘P’.

Triple integrals

Double integration is used to integrate over a two dimensional region, therefore a triple integral is used to integrate over a three dimensional region.

 $\int\int_{A}\int f(x,y,z) dv$

Where, A = [a, b] x [c, d] x [e, f]

That is  = $\int \int_{A} \int f(x,y,z)dv$ = $\int_{e}^{f}\int_{c}^{d}\int_{a}^{b}f(x,y,z)dx dy dz$

Here we first integrate the function with respect to ‘x’ then with respect to ‘y’ and then with respect to ‘z’

Application of Integrals

Integration has a large number of applications in science engineering. Integrals can be used to find
(i)    Area under a curve
(ii)    Area between two curves
(iii)   Volumes of solids with known cross sections
(iv)   Volumes of solids of revolution ( using disks and washers)
(v)    Arc length
(vi)   Area and volumes involving parametrically defined functions
(vii)   Area and arc length for polar curves

Fundamental Theorem of Calculus

The first fundamental Theorem of Integral calculus

If ‘f’ be a continuous function of ‘x’ defined in the closed interval [a, b] and the function F is an anti derivative of ‘f’ on [a, b], such that F’ ( x ) = f( x ) for all ‘x’ in the domain of ‘f’, then $\int_{a}^{b} f(x) dx$ = [F ( x ) + C ]ba

        = F ( b ) + C – [F ( a ) + C]

        = F( b ) + C – F( a ) – C

        = F( b )  – F( a )

The Second fundamental theorem of calculus

If ‘f’ is a continuous function on [a, b] then

$\frac{d}{dx} \int_{x}^{a} f(t) dt $=f(x)

Numerical Integration

Sometimes it is extremely difficult or even impossible to evaluate a definite integral $\int_{a}^{b} f(x) dx$ even if f( x ) is continuous in [a, b]. In such cases we take a set of numerical values of the integrand ‘f’ in the interval [a, b] and evaluate the definite integral $\int_{a}^{b} f(x) dx$ approximately. This process of finding the approximate value of a definite integral is called Numerical Integration. If the integrand is a function of single variable, then this process is known as Quadrature.

In Numerical Integration first we approximate the integrand ‘f’ by a polynomial ‘p’ and then find $\int_{a}^{b} p(x) dx$
Then $\int_{a}^{b} f(x) dx$ is approximately equal to $\int_{a}^{b} p(x) dx$. The absolute difference | $\int_{a}^{b} f(x) dx-\int_{a}^{b} p(x) dx$| is called the error term in such approximation. There are several methods for approximating a definite integral, such as Trapezoidal Rule, Simpson’s Rule etc.

Trapezoidal Rule

Let f: [a, b] → [0, ∞ ) be continuous and P = {a = x0, x1, . . . . . . . . . . xn= b} be a partition of [a, b] into ‘n’ equal sub intervals, each of length h = $\frac{(b - a)}{n}$

Then, $\int_{a}^{b} f(x) dx$ =$(\frac{h}{2})$(y0 + y1) + $(\frac{h}{2})$(y1 + y2) + $(\frac{h}{2})$(y2 + y3) + . . . .  . . . . . . .  . . . . + $(\frac{h}{2})$(yn - 1 + yn)

 $\int_{a}^{b} f(x) dx$ = $(\frac{h}{2})$* [(y0 + yn) + 2(y1 + y2 + y3 + . . . . . . . . . . . . .. . . + yn - 2)]

Working Rule for Trapezoidal Rule

Step 1: Note down a, b, and n the number of divisions

Step 2: Calculate the value h =$\frac{(b-a)}{n}$

Step 3: Compute the values of ‘f’ at x = x0 ( = a), x = x1, x = x2, . . . . . . . . . . . .  x = xn = b and denote them as y0, y1, y2 . . . . .. . . . . . . . . . . yn respectively.

Step 4: Write down the Trapezoidal rule for the division specified in the question and substitute the values obtained in step 3 in the rule 91) and simplify.