Sub:- Applied Mathematics-II Topic: Integral Calculus-I Sub:- Applied Mathematics-II Topic: Integral Calculus-I By:-MATH FACULTIES

Introduction:- Integration literally means “adding small parts to make up the whole”. In integration we get the function whose rate of change is known i.e. integration is inverse process of differentiation.

The symbol dx indicates that the integration is to be performed with respect to the variable x For example:-

Some Standards Results:- xn dx= (xn-1 / n+1) + c, where c is constant of integration. Here n -1 2. (ax + b)n dx = n+1, n -1 4.

ax dx= ax / log a ex dx = ex Cosx dx = Sinx 8. sin x dx = -cos x 9. tan x dx = -log x 10. cot x dx = log sin x

Sec2 x dx = tan x 12. CoSec2 x dx = -Cot x

Sec x tan x = Sec x Cosec x Cot x = - Cosec x Sec x dx = log (Sec x + Tan x) Cosec x dx = log (Cosec x – Cot x)

Some Results:- where c is constant & f(x) is. a function.

For example:-

Integration by Substitution Method:- When ever a function f(x) occurs in the integred along with its exact derivative f’(x) or a constant multiple of the same, than the substitution f(x)=2 reduces the integred to integrable form. i.e. we put f(x)=z, we get f’(x) dx =dz

Two Standard Forms:-

Some Standard Forms:-

Integration by parts:- where u and v are functions.

Integration by Partial fractions Method In this method given rational algebraic fraction is split into a number of fractions and each fraction is integrated.

APPLICATION OF INTEGRATION AREA AND SIMPSON’S AND TRAPEZOIDAL RULE

Sub :- Maths (Integral Calculus) Topic- Definite Integral

Introduction:- Today, in this topic we will learn to evaluate definite integral of different functions using properties of Definite Integral.

TOPIC : STATISTICS Mean Median Mode Standard Deviation

What is statistics? a branch of mathematics that provides techniques to analyze whether or not your data is significant (meaningful) Statistical applications are based on probability statements Nothing is “proved” with statistics Statistics are reported Statistics report the probability that similar results would occur if you repeated the experiment

Statistics deals with numbers Need to know nature of numbers collected Continuous variables: type of numbers associated with measuring or weighing; any value in a continuous interval of measurement. Examples: Weight of students, height of plants, time to flowering Discrete variables: type of numbers that are counted or categorical Numbers of boys, girls, insects, plants

Statistical Computations (the Math) If you are using a sample population Arithmetic Mean (average) The sum of all the scores divided by the total number of scores. The mean shows that ½ the members of the pop fall on either side of an estimated value: mean

Median: the middle number Mode and Median Mode: most frequently seen value (if no numbers repeat then the mode = 0) Median: the middle number If you have an odd number of data then the median is the value in the middle of the set If you have an even number of data then the median is the average between the two middle values in the set.

Standard Deviation An important statistic that is also used to measure variation in biased samples. S is the symbol for standard deviation Calculated by taking the square root of the variance So from the previous example of pea plants: The square root of 2.5 ; s=1.6 Which means the measurements vary plus or minus +/- 1.6 cm from the mean

What does “S” mean? We can predict the probability of finding a pea plant at a predicted height… the probability of finding a pea plant above 12.8 cm or below 3.2 cm is less than 1% S is a valuable tool because it reveals predicted limits of finding a particular value