 next is the property number six which is called the even odd property you'll find a lot of resemblance between property number four sorry property number five and property number six which i'm going to do right now it says that if you are integrating a function from minus a to a it becomes integration of f of x from zero to a plus integration of f of minus x from zero to a let me prove this mathematically first yeah yeah so keert i got your message okay so let me prove this mathematically let me start with our left hand side so minus a to a f of x you can break it up as minus a to zero and zero to a let's not disturb this part let's work on this part okay now looking at the proof looking at the result only you'll get the idea how to prove it you just have to put x is minus t correct so this becomes dx becomes minus of dt the limits of integration will become a to zero now so f of minus t dx is minus dt you can switch your position of the upper and the lower limit and observe the negative sign by the way right hand side will be just carry forward so here just switch the upper and the lower limit and observe the minus sign and finally there's nothing in the name you can put your t back as x so zero to a f of minus x dx plus zero to a f of x dx this is what the property says not two important conclusions that we can draw from here i'll write down the conclusions here conclusion number one if your f of x is even function if your f of x is an even function that is to say that f of minus x is f of x also has to say that the graph is symmetrical about x equal to zero line which is nothing but your y axis then your integrant from minus a to a f of x dx is going to be two times zero to a f of x guys remember i have not said this but the function is continuous from minus a to a conclusion number two is this rotation symmetry than zero yes if f of x is an odd function that means f of minus x is negative of f of x that means the graph is symmetrical about zero comma zero then what will happen integral from minus a to a f of x dx will become zero okay it's very obvious that if a function is an even function what will happen the graph will be exactly symmetrical about x equal to zero line which is your y axis so instead of finding this area you can just find out this area and double it up okay so just this area you double it up your job is going to be done so this into two will be your total area but when your function is symmetrical about origin then that means there is a cancellation of area happening let's say i have a graph like this okay this area and this area will exactly cancel each other out so integral from minus a to a will vanish okay so this area will be zero now this is a very important conclusion because we know that every function can be written as a sum of an even and an odd function correct we have done this in our functions chapter so whenever you see that your limits of integration are exactly opposite in sign try to extract out as many odd functions as you can from your given function and just make it zero the integral of that will be zero are you getting my point and you can only work with the integral of the even function that is going to save a lot of time for you let's take a quick question on this sir is it true okay shrijan says two sir no sir suki also says two okay the no sir i only oh okay okay even he said okay privately he said two okay let's check this out this is quite easy you just have to segregate the odd functions odd functions will be yes or two yeah odd functions will be everything except that one so that one you have to take separately and treat it as as per the even function criteria okay so this you don't have to evaluate at all because this is an odd function okay so this is just nothing but twice zero to pi by four integration of secant square x that's nothing but twice tan x zero to pi by four answer is two let's take the next one suppose a function g n x is given by this and it satisfies this equation integral minus one to this is equal to zero for all linear function px plus q then which of the following option is correct sir if it satisfies well all linear functions then one g n x be zero like zero function why sir because if you consider the areas then it should be like that right because px plus q can be anything it can be like any line so so the areas will be it will be unequal so if you want it to become zero then i thought it should be give me a better easy b that's correct shrija is correct anybody else got a value shouldn't be a problem actually yes sir it's easy it's very easy yeah that means this should be completely an odd function correct now what odd function now what are the even parts here even parts i can see is px to the part 2n plus 2 okay that's also pan x square and it is bn cube correct these are the even uh sorry even part of the function yes or no odd part i don't have to worry much about odd part because odd part anyways integral is going to be giving me zero so what they're saying that even the integral of these parts from minus one to one should give you zero correct one thing that i can say here for sure that this thing should be zero that means bn has to be zero for all q so whichever option has bn equal to zero only those option can qualify okay so a and b could be the possible options c and d cannot be okay next if you integrate this you get p x to the power 2n plus 3 by 2n plus 3 and this you get pan x to the power 3 by 3 okay twice of this sorry i've already integrated it so i have to just put zero to one okay so it's p by 2n plus 3 is equal plus pan by 3 that's going to be zero right now p need not be zero that means if you take p common or if you directly drop off your p remembering p is not zero so your an will be negative 3 by 2n plus 3 negative 3 by 2n plus 3 which is option number b is what it is saying b is the right option okay so a bit of analysis has to go in next question if your function is given by this determinant then find the integral from minus pi by 2 to pi by 2 of x square plus 1 f of x plus f double dash x anyone it's almost done sir one please also see the video lecture of the definite integral session taken with the other batch so there are done more basic problems here i am taking slightly more challenging ones so but also remember to see then see those videos once your exam is over okay no hurry after the exam is over you can watch those videos let's see who has responded so keep your correct see guys it's it's just a one minute problem one minute not more than that if you look at the function if you replace your x with minus x okay remember nothing will change in the first column nothing will change in the second column but everything in the third column will become negative isn't it okay which clearly implies this is negative of f of x that means f of x is an odd function correct this clearly implies f of x is an odd function okay one sorry okay thank god okay so even the double derivative will also be an odd function correct which implies the sum of these two will also be an odd function correct now remember x square plus one is an even function x square plus one is an even function okay so x square plus one into f of x plus f double dash x is a product of even and odd which will actually be odd so if you're integrating an odd function from minus pi by 2 to pi by 2 your answer has to be zero which is none of these zero is the answer so no need to evaluate the determinant at all it was just based on your understanding of the basic property they're actually evaluated oh my god then also but then I cancelled off most of the things because it was odd so this is a type of learning for you okay hope you can read this question slightly blurred let me just scribble on it wait how do you show that even the double derivative is also an odd function if at all f of x is odd the derivative of an even function is odd correct and derivative of an odd function is even where I didn't know that function chapter we did this no oh yeah yeah I hope you can read this question now this is normal brackets so it's it's nothing but integral of x plus one by x minus one square plus x minus one by x plus one square minus two whole to the power half from minus half to one zero no god of that's not zero so one second two let's check let's check okay now I'm sure all of you would have two okay let's check most of you would have figured out that this is actually x plus one by x minus one minus x minus one by x plus one whole square under root correct which actually is mod of x plus one by x minus one minus x minus one by x plus one okay I'm not sure how many of you accounted for this mod thing many of us have this habit of writing square under root as the same thing which is not correct it's actually mod of the same thing for my mistake sir okay yeah umkar seems to be correct let's just discuss it completely so this will become minus half to half if you take the lcm inside it will become x square minus one yes four x four x will come up correct now if you draw the wavy curve for four x by x square minus one which is actually four x by x plus one x minus one the wavy curve will give you the sign scheme like this correct now you're dealing with minus half to half minus half is here and half is here correct so can I say from by the way even even I don't need to do that I can say that this function will always be an even function this entire function is an even function why because if you change your x with minus x it just becomes minus four x by x square minus one mod which is same as mod four x by x square minus one that means f of minus x is f of x itself so I can first make it as two times zero to half mod four x by x square minus one next four minus one s t and now I don't even need this part I can only deal with zero zero to half interval where the function is negative so it becomes negative four x by x square minus one I'm sure by this time you would have known how to integrate this so if you take a minus two out so minus four zero to half two x by x square minus one this integral is minus four ln x square minus okay put first half half will give you ln three by four okay zero will give you zero so that is just going to be minus four sorry plus four ln four by three okay sir isn't ln of minus or not defined no you're taking mod okay okay okay so I think this is your answer sir I didn't get it actually I miss my constant I think yeah I miss my constant yeah yeah is that fine guys this chapter is pretty long it's not a small chapter we have a lot of things to cover I think the next class which is next class on Thursday Friday is your English exam right lol yeah so please study English before and don't bunk on that day that's going to be an important class sorry we won't bunk it but we won't study English before us exactly I know last moment you guys are going to study anyways we'll do a lot of important things the next class so as of now I'm just stopping the session over here and all the best for tomorrow's exam thank you sir thanks thank you thank you sir bye bye have a good day thank you sir