Integration By Parts Formula and Mean Median Mode Formula

Integration by Parts Formula

Integration is a very important computation of calculus mathematics. Many rules and formulas are used to get integration of some functions. A special rule, which is integration by parts, is available for integrating the products of two functions. This topic will derive and illustrate this rule which is Integration by parts formula. Also, some examples will help the students to get their concept. Let us start!

Integration by Parts Formula

What is integration by parts method?

This method is very useful in order to master the technique of integrations. Many times we have to integrate the product of two functions. Functions often arise as the products of other functions, and so we have to integrate these products. For example, we may be asked to determine

∫x; cosx; dx

Here, the integrand is the product of the two functions x and cos x. A rule exists for integrating the products of functions which is required for getting the solution.

Derivation of the formula for integration by parts:

This rule states that:

Undefined control sequence \( dx = uv – \int {\frac{{du}}{{dx}}} vdx \)

Derivation: If y = uv

As we know that,

dydx= ddx uv = u ddx v + v ddx u

Rearranging it,

uddxv=ddxuv–vddxu

Now integrating both sides,

∫udvdxdx=∫d(uv)dxdx–∫vdudxdx

The first term on the right hand side simplifies since we are simply integrating what has been differentiated.

∫udvdxdx=uv–∫vdudxdx

This formula is known as integration by parts. This formula is very useful for solving complex integration problems. In some questions, we will see that it is sometimes necessary to apply the formula for integration by parts more than once.

This gives us the method for integration, called INTEGRATION BY PARTS. This method allows us to integrate many products of functions of x. We take one factor in the product as u (this also appears on the right-hand side, along with dudx).

The other factor is taken to be dvdx  (on the right-hand-side the only v appears)

Solved Examples

Q: Solve the integration given:

∫xcosxdx

Solution:

Here, we are trying to integrate the product of two functions x and cos x. To use the integration

by parts method we let one of the terms be

dvdx and the other be u.

See from the formula that whichever term we let equal u we need to differentiate it in order to find dudx

So in this case, if we assume u as x, so when we differentiate it we will find

dudx==1,

i.e. simply a constant. Notice that the formula will replace one integral, the one on the left, by another, and one on the right. Careful selection of u will produce an integral which is less complicated than the original.

Thus

u=xanddvdx=cosx

With this choice, by differentiating we obtain

dudx=1

Also from dvdx=cosx,byintegratingwefind

v=∫cosxdx=sinx

Then use the formula,

∫udvdxdx=uv–∫dudxvdx

∫xcosxdx=xsinx–∫(sinx).1dx

=xsinx+cosx+c , which is the solution

Where c is constant of integration.

Mean Median Mode Formula

In our day to day life, we often hear that the average height of students of class 10 is 5 ft. and 5 inches or the average marks of 12th Board exams were 75%, etc. But have you ever wondered how did this average get calculated? It is with the help of the Measures of Central Tendency. A Measure of Central Tendency refers to a single or individual value that defines the manner in which a group of data assembles around a central value. In other words, one single value describes the behavior pattern of the whole data group. Let’s start learning the three measures of central tendency i.e. Mean Median Mode Formula.

Mean Median Mode Formula

 What is Mean?

The mean of a series of data is the value equal to the sum of the values of all the observations divided by the number of observations. It is the most commonly used measure of central tendency. Also, it is very easy to calculate. We denote Mean by X¯¯¯¯.

Mean Formula

1. Individual Series

X¯¯¯¯=SumofallthevaluesoftheobservationsNo.ofobservations

2. Discrete Series

a. Direct Method:

X¯¯¯¯=∑fx∑f

Where,

f	Frequency
x	Values

b. Assumed Mean or Short-Cut Method:

X¯¯¯¯=A+∑fd∑f

Where,

f	Frequency
d	X – A
A	Assumed Mean

 c. Step-Deviation Method:

X¯¯¯¯=A+∑fd′∑f×C

Where,

f	Frequency
A	Assumed Mean
d’	(X–A)C
C	Common factor

3. Frequency Distribution or Continuous Series:
4. Direct Method:

X¯¯¯¯=∑fm∑f

Where,

f	Frequency
m	Mid-Values

1. Assumed Mean or Short-Cut Method:

X¯¯¯¯=A+∑fd∑f

Where,

f	Frequency
d	m – A
A	Assumed Mean

1. Step-Deviation Method:

X¯¯¯¯=A+∑fd′∑f×C

Where,

f	Frequency
A	Assumed Mean
d’	(m–A)C
C	Common factor

Weighted Arithmetic Mean Formula:

X¯¯¯¯=∑WX∑W

Where,

X	Values
W	Weights

What is the Median?

Median is the central or the middle value of a data series. In other words, it is the mid value of a series that divides it into two parts such that one half of the series has the values greater than the Median whereas the other half has values lower than the Median. For the calculation of Median, we need to arrange the data series either in ascending order or descending order.

1. Individual Series

2. When the number observations are odd

M=Sizeof(N+1)2thterm

Where,

N	Number of observations

1. When the number observations are even:

M={Sizeof(N+1)2thitem+Sizeof(N2+1)thitem}2

Where,

N	Number of observations

2. Discrete Series:

Median=Sizeof(N+1)2thterm

Where,

N

∑f

In this case, the value corresponding to the cumulative frequency just greater than the value obtained after applying the above formula is the Median of the series.

3. Frequency Distribution or Continuous Series:

Firstly, we need to calculate the Median class by applying the following formula:

Median class = N2

M=l2+hf[N2–c.f.]

Where,

l	Lower limit of the Median class
h	Size of the median class
f	Frequency of the median class
N	Sum of frequencies
c.f.	Cumulative frequency of the class just preceding the median class

What is Mode?

Mode refers to the value that occurs a most or the maximum number of times in a data series.

Mode formula

1. Individual Series:

We find the mode of an individual series by simply inspecting it and finding the item that occurs maximum number of times.

2. Discrete Series:

The Mode of a discrete series is the value of the item that has the highest frequency.

3. Frequency Distribution or Continuous Series:

Firstly, we need to find out the Modal class. Modal class is the class with the highest frequency. Then we apply the following formula for calculating the mode:

Mode = l +  h    f1−f0(2f1−f0−f2)

Where,

L	lower limit of the modal class
f1	Frequency of the modal class
f0	Frequency of the class just preceding the modal class
f2	Frequency of the class just succeeding the modal class

Solved Example

1. Calculate the Average marks of the following series using Direct Method.

Marks	No. of students
0 – 10	10
10 – 20	30
20 – 30	70
30 – 40	50
40 – 50	20

Solution:

Marks	Mid-values (m)	No. of students (f)	fm
0 – 10	5	10	50
10 – 20	15	30	450
20 – 30	25	70	1750
30 – 40	35	50	1750
40 – 50	45	20	900
		∑f=180	∑fm=4900

X¯¯¯¯=∑fm∑f

4900180

= 27.22