Table of Integrals

Integration is a fundamental operation in integral calculus. Integration means calculating the area made by curve. While doing integration, table of known integral are very useful. There are various types of table of integral. First type is definite integral table, definite integral means integration of function with definite limit. Such as integration of fx dx where limit are x=a and x=b, which made an area. Suppose limits are indefinite then it is known as improper integral. For this type of problem we use suitable limiting procedure to convert indefinite limit to definite limit.Definite integral table contain many expressions. In contains elliptic integral, square root, arc tangent means inverse tangent functions, exotic function and some special functions.

Gaussian integral table contains expressions of erf function or error function, Gaussian integral table also known as probability integral.  In this type of integration  we have to integrate one dimensional Gaussian function over the limit from negative infinite to positive infinite(-ve8, +ve 8) it can be solve by using a technique like combining to, one dimensional function in Gaussian function. Here we integrate first dummy variable present in the integral and carryout the term in the end. The n it becomes function of one variable. Now we move to polar coordinate. It is not necessary to use polar coordinates, also we can proved in simple way. Now we take continued function whit erf, or error function.  To solve this function we use Laplas method.

Exponential integral table contain integration of exponential functions. Exponential functions means e^x functions, where e is a number whose value is 2.718. Integration of exponential functions means constant change in the independent variable gives same change that is proportional to input. Exponential functions some time written as exp(x), but it is unpractical to write for any independent variable. Other than mathematics exponential functions are also used in physics and chemistry. Some integral table exponential function are-

Exponential function = e^x
Inverse exponential function = ln x
Derivative of exponential function = e^x
Indefinite integral of exponential function = e^x +p

Increasing exponential function always above from x axis and closed to negative value of x. some time it can be expressed as cbx in this form base b is real number, Variable x can be real or complex number and c is a constant term. Decreasing exponential functions always below from x axis and closed to positive value of x.

Prime Numbers

Prime Numbers:

A prime number is one that has only two factors namely 1 and itself and a composite number has factors besides 1 and itself.


A natural number greater than 1 that has no divisor between 1 and itself is said to be prime, hence called a prime number or simply a prime. Every natural number greater than 1 has at least the two distinct divisors 1 and itself; a prime has no others.








The number 2 is a prime, there being no candidate divisors between 1 and itself; from it, all even numbers thereafter are non-prime, i.e. 50% of all subsequent numbers. The numbers 3, 5, and 7 are all prime, meaning that, of the first six such subsequent numbers, precisely half are prime, half non-prime. However, of any subsequent six consecutive numbers, at least one of the odd values must be divisible by 3; including the three even numbers this means that at least 66% must be non-prime. So the trend goes; as we look further afield, with an accumulating collection of primes to be divisors, the density of primes declines progressively. But, no matter how far up the numbers we travel, we never exhaust the primes, nor is there any known point above which all further primes are spaced by more than the minimal value of 2 .

Hope you like the above example of Prime Numbers.Please leave your comments, if you have any doubts.

Finding variance and variance analysis

We know that the mean is a measure of central tendency. However, what we don’t know is that the mean is many a times not very useful in making informed decisions. Let us consider the following example to drive the point.Suppose one cricketer is to be selected from two on the basis of his batting performance.

The scores of the cricketers A and B in the last 5 innings which they played together are as follows:
Cricketer A: 51, 53, 52, 55, 59
Cricketer B: 85, 23, 69, 07, 96

If we compare the scores based on the mean, then mean of cricketer A is (51 + 53 + 52 + 55 + 59)/5
= 54 runs
And the mean of cricketer B = (85 + 23 + 69 + 07 + 96)/5 = 56 runs.

Thus if the decision is to be taken on the basis of comparison of the means only, then cricketer B would be selected since he has a higher mean runs. But if we take into consideration the reliability of A and B we can obviously notice that B is not reliable, because he is not consistent. B scores 96 runs in one inning but at the same time scores only 7 runs in another inning also. A is consistent in his score and so is more reliable. In other words the difference between two scores of B is very large where as in case of A it is very small. It means that the data regarding the scores of B deviates from the centre instead of concentrating at the centre.

This tendency of a data set to deviate from the mean is called dispersion and its value is called the measure of dispersion. Thus to come to a sensible conclusion from a given data set, we need to know the mean as well as the dispersion. The lesser the dispersion, the more reliable is the mean of the data.

Define Variance: Variance is a measure of dispersion from the mean. Just like how we saw in the example above, variance analysis helps us to understand dispersion of data from the mean so that we can decide better.

Formula for variance: For finding variance we use the following formula:
Variance = V =[ (X1-M)^2 + (X2-M)^2 + (X3-M)^2 + …. + (Xn-M)^2]/n
Where, X1,X2,X3,….Xn are the n observations and M is the mean of these n observations.

Second Order Differential Equation

A second order differential equation is an equation involving the unknown function y, its derivatives y’ and y” and the variable x. in the mathematical form we can write d^2y/dx^2= f(x,y,y’).  This equation is the basic form of second order differential equation.

Let’s discuss some types of second order differential, first is second order linear differential equation. The most general form of second order linear differential is p (t) y”+q(t)y’+r(t)y=g(t).  Where p,q,r and g are continuous function, we can use this equation to study of the motion of spring. In the above equation if g(t) =0, the its is called homogenous linear equation so the form of second order linear homogeneous equation is p(t)y” + q(t)y’ + r(t)y = 0. As same if g(t) is not equal to 0 then the equation is called nonhomogeneous linear equation.

Now lets consider second order linear homogenous equation, now to solve this type of equation, there are two basic facts that we have to consider to solve linear homogeneous equation. The first fact says that if we know two solution y1(x) and y2(x) of such a equation, then the linear combination y= c1y1(x) +c2y2(x) is also a solution. The second facts says that if y1 and y2 are linearly independent solution of such equation (second order linear differential equation) and p(t) is never 0, then the general solution is given by y(x) = c1y1(x) + c2y2(x).

Generally this is not very easy to find the particular solution to second order linear differential equation, but it is always possible to do so if the coefficient p, q, and r are constant functions, that is, if the differential equation has the form
ay” + by’ + cy = 0, where a, b and c are constant and a is not equal to 0.

There is also some Second Order Differential Equation Solver, you can find it online. We can use MATLAB to solve second order differential, even we can find plot for the equation. In MATLAB we can solve second order ordinary differential equation with out any knowledge of numerical methods. MATLAB has number of tools to solve ordinary differential equation, but the two main tools are ode23 and ode45 which implement the version of RUNGA KUTTA 2nd/3rd and RUNGA KUTTA 4th/5th respectively. In MATLAB we can use LAPLAS transform, EULER’S method   and Taylor methods.  MATLAB has some additional solvers like ode113, ode23s, ode15s, ode23t and etc.

Stepwise Regression

Regression, in statistics regression means modeling and analysis of variables. It shows the relationship between a dependent variable and a independent variable, it may be one or more then one.  Now we more simplify the term regression, it shows how a dependent variable changes when we vary one independent variable and treat another independent variable as a constant. Regression is used for prediction and forecasting. It also shows how an independent variable is related to dependent variable.

What is stepwise regression?
Stepwise regression means regression analysis in which the choice of predictive variable is taken out by an automatic procedure. If we more specify step-wise regression then suppose we have any mathematical equation with several dependent and independent variables which we have to solve by step-wise regression modeling.  By applying several techniques we first eliminate the independent variables one by one to find the final value.

For better understand the step-wise regression we have to take the stepwise regression example, because with example we easily understand the procedures. For this we take a quadratic equation such as (aX^2+bX+c). In this we have to find dependent variable X by eliminating the independent variables. In step one; we separate the middle term in two parts with checking suitable multiplication of first and third term and addition also. In step two, we take out the common terms. In step three, we equate each term with zero and then finally we get the value of X.

Step-wise regression is of various types. First we discus about forward step-wise regression, this type of regression starts with no variables in the model.  Basically we have to improve the model in forward stepwise regression by applying suitable procedure, such ass apply addition of each variable from a given model and improve the model. Repeat the procedure until none improvement take place. The total calculation is based on the F-distribution and degrees in the regression terms.

Now we discuss about another type of stepwise regression which is stepwise linear regression. This type of regression shows the relationship between a dependent variable which is scalar and one or more explanatory variables. If one explanatory variable is used then simple regression and more than one explanatory variable is known as multivariate linear regression, where the multiple variables are predicted which correlated with dependent variable. In linear regression linear predictor functions are used in regression analysis. Linear regression has also many practical uses.

Limits of Infinity Simplified

In Mathematics, one of the mysteries is the concept of ‘infinity’ which represents very large numbers.  We come across +infinity and –infinity. 1 divided by zero gives us + or – infinity. To be precise 1/0+ gives us +infinity and 1/0- gives us –infinity. But 1/infinity is undefined as we cannot determine the value of it.  Let us consider a variable x and arrive to the value of 1/x. When we plug in large values of x, the value of 1/x gets smaller and smaller. We can conclude that as x gets larger, 1/x tends to zero but does not equal zero.  To say that the value equals zero we use the term limit in mathematics, using which we can say, as the value of x approaches infinity, limit of 1/x equals zero. Here is where infinite limits come into picture, they are some functions which increase or decrease without bounds near certain values for the independent variable. Infinite limits are represented as , lim (x->-infinity) a/x^r= zero and lim(x->+infinity) a/x^r= zero for any number a, r being a positive integer and x^r is a defined function.  We also have lim(x->infinity)x^r= infinity and for any natural number ‘n’, lim(x->-infinity)x^n=(-1)^n infinity.

Let us take a look at some of the limits at infinity examples, evaluate lim(x->infinity) [x^2+1]/[2x-3]. Let us re-write the given function, (x^2/x).[1+1/x^2]/[2-3/x] which gives us x.[1+1/x^2]/[2-3/x]. Applying limit we get, lim(x->infinity)[1+1/x^2]/[2-3/x] gives us (1/2)x  and 1/2 lim(x->infinity)x= infinity

Finding infinite limits of a given function, infinite limits are given by lim(x->a) f(x) = +infinity or lim(x->a) f(x)= - infinity. Solving Infinite limits, for instance let us solve lim(x->0-)[x^2-1/x]. We can see this limit is going to infinity, here we need to determine whether it is positive or negative infinity. Simplifying [x^2-1/x] we get [x^3/x – 1/x] which gives (x^3 -1)/x. Now substituting (-0.1) for x. we get, [(-0.1)^3-1]/(-0.1)= negative/negative  which gives positive and hence we get positive infinity as the answer.

Infinite limits are defined as lim(x->a) f(x)= infinity  for all x close to x=a, from both sides, but not letting x=a and limit(x->a) f(x)= - infinity for all x close to x=a, from both sides, but not letting x=a. Evaluating infinite Limits, lim(x->0-)3/x^2. As we can see, it is a left hand limit. In this case, we take smaller and smaller values of x and when we square them, they will get smaller, but on squaring the values will be positive. So, now we have a constant divided by an increasingly small positive number which gives a positive infinity, that is lim(x->0-)3/x^2=positive infinity.

Limits at Infinity

Limit of a function at a point may or may not exist. If limit of a function is not exist then may be +  or - . If limit of a sequence does not exist then it is called as a divergent sequence. It may be divergent to +  or divergent to - .Limits at Infinity has two possibilities one is positive side and other is negative side. For some sequences there may not exist finite or infinite limit values. 

Infinite Limits are the limits for which limit value does not exist. When we are finding limit for the division of two functions if it does not exist then we can use L-Hospital rule to find out the limit value. L-Hospital rule states that, if we are getting

form when we are finding limit for division of two functions then first differentiate numerator and denominator and then apply limit.