Hi friends so today we are gonna learn a new concept called as beta function now beta is a Greek symbol which is denoted like this or capital B so beta is a function of we can say let’s M and N which is defined by a definite integral from 0 to 1 X raise to M minus 1 1 minus X raise to N minus 1 DX so this is the definition of beta function so if you’ll observe here then if we have the value M then inside the integration we get power as M minus 1 that if this power is always 1 less than this M so if I say that if this power of X is M then definitely inside this B or beta we are going to get M plus 1 and if this power is n then we are going to get n plus 1 so this is also the same definition so this form we will get in many problems where we are going to substitute it as beta of M plus 1 and n plus 1 similarly we have one more definition of beta function in terms of integration which is given as integration from 0 to infinity X raised to M minus 1 upon 1 plus X raise to n plus 1 DX so this definition is also helpful to solve certain problems where the integration can be from 0 to infinity or it will be in such format so these two definitions are in the algebraic form if you will see the right hand side we have algebraic terms over here now similarly we can find out the second form of beta function by doing the particular substitutions in the first equation so second form of beta is given in terms of trigonometric functions so if I’ll substitute or if I put X as sine square theta in the first definition then I can get the second form of beta function so let us see what is the second form so if we put X equal to sine square theta then DX would be the derivative of this is two sine theta cos theta d-theta similarly the limits will change as following so X will be converted into theta for X we have limit from zero to one now when x is 0 this will become sine inverse of zero hence the theta is also 0 and when X is 1 sine inverse of 1 will become PI by 2 hence the limits are changing from 0 to PI by 2 now we’ll substitute these values in the equation number 1 so that will get equation number 1 as beta M comma n as integration 0 to PI by 2 here X which is sine square theta will give us sine of 2m minus 2 theta it is sine raised to 2m minus 2 theta 1 minus X that is 1 minus sine square theta which is cos square theta and cos square theta raised to n minus 1 will give us cos raised to 2 n minus 2 theta similarly this DX will be substituted by 2 sine theta cos theta d-theta so that will be 2 sine theta cos theta d-theta now if we will solve this I can take this 2 outside integration 0 to PI by 2 this sine term has power 1 if I multiply this with sine raised to 2m minus 2 it will give me sine raised to 2 N minus 1 theta similarly this COS will give me cos raise to 2n minus 1 theta d-theta and this is second form of beta function so I will say this as equation number three so this form is useful to solve this some where the question is given in terms of trigonometric functions so whenever we have the integration which has only trigonometric terms that time we are going to use the second form of beta function so after this let us see certain properties of beta function which can be useful to solve the problems based on beta function so out of that the property number one is beta of M comma n is always equal to beta of n comma M now we can easily prove this so this property is useful in certain problems let us see property number two it is now property number two is basically the relationship between beta and gamma function so the relations is beta of M comma N is nothing but gamma M gamma and upon gamma M plus n so this property is useful for us to solve many problems based on beta function so whenever we want to evaluate the beta we are going to use this second property because we know how to evaluate the gamma function hence instead of using integration every time we evaluate the value of beta M comma n we are going to use the second property so that we can get answer in less number of steps now after this we will see the second form of beta function with the different powers now let us say that this 2m minus 1 is equal to P and this 2n minus 1 is equal to Q so if I put 2 m minus 1 H P and 2 n minus 1 sq then I can get the value of M as p+ 1 upon 2 similarly value of n can be Q plus 1 upon 2 now if we substitute these two values inside the second form we can get third property as 2 times integration from 0 to PI by 2 sine raised to P theta cos raise to Q theta D theta as beta of P plus 1 by 2 Q plus 1 by 2 so this property again we are going to use whenever we have the sums based on trigonometric function so we will use these properties and definition to solve many problems of beta function thank you

Nice video! I like it

Nice video! I like it

good explaination ………………..

Excellent! Thanks.

nice explanation

extrodinary work Sir….Hats off to your effort!!!

I have a doubt, in equation no. 2, the term in denominator has power written as (n+1). So is it (n+1) or (m+n) because in the book it says (m+n). Please reply ASAP.

thank you

Nice explanation

What is the property for Z π/2

0

tana x

cota x

dx =

sir thanks u r tooo good and your way to make us understand the difficult problems is very good

Sir I want whole m2 subject videos

Suruwat me hi 2nd defination me x^m-1/ {1+x}^ m+n aayega

Basic hai bhai

nice video sir thank you so much…

the way of teaching was very convenient and easily understandable.

Sir can you please post vedio on Gauss divergence theorem and stokes theorem

Sir khub bhalo bujhieye che apni… Very good sir

Kintu sir aapnar chuler colour green mone hoche… Jani na keno..

thankyou sir😍

How can i got ur all lectures

👌👌👌👌👌👌

Clear concept

Denominator should be m+n not n+1

Superb👏

You have written the second equation wrong involving a wrong power of the (1+x) term. It should be m+n. Why do you come here to misguide those who don't know. It's better to not know than learning wrong things. Do you homework well before even trying to teach others. You need to be very careful. Don't you check even once before presenting your lectures? Do you think or prepare yourself? If it were some careless student, they would just memorise and use the wrong expression and end up losing marks or teaching the same wrong thing to others. #shame

Thanks ser

Waaaait

From 2:55 onwards –

Sin is squared. How would it become sin inverse? Won't it become (sin^-2)x?

Hello Friends,

Watch Complete Video Series of Subject Engineering Mathematics 2 only on Ekeeda Application.

Use Coupon Code "NEWUSER" and access any one Subject free for 3 days!

Download Ekeeda Application and take advantage of this offer.

Android:- https://play.google.com/store/apps/details?id=student.ekeeda.com.ekeeda_student&hl=en

iOS:- https://itunes.apple.com/tt/app/ekeeda/id1442131224

Nice…

Useful

Hindi nahi aate Kay