DIPLOMA MATHS-M3 MID-1||C-24||ASK ENGINEER ACADEMY||

[00:00:00]indefinite meaning there is no limit.

[00:00:02]That is the reason a toe to fx definite form of integration integration integration a b x dx type of integration definite form of integration definite definite definite form of The form of integration term. Form of integration term definite form of integration suppose to integration limits f of x same dx is there. So there is no limits that is called indefinite indefinite indefinite indefinite form form of integration form of integration beta. So indefinite form of integrations first try to understand form of integration.

[00:01:09]Next only integration limitation.

[00:01:25]The same procedure will be followed for the definite form of integrations.

[00:01:31]Definite only a to some of the properties of the definite form of integration we will going to be studied here right. So first integration. So this is about the next integration process. This is what is integration. Integration it is the summation process. Integration is called it as a it's a summation process. It's the first definition.

[00:02:14]Otherwise integration is the continuous summation process. Okay. Continuous summation process.

[00:02:27]amount of values that is called summation. Integrating means sum summation of the process that is called summation process. This the first thing and second point is here very very important point this integration is the anti-derivative anti-derivative. So what is this anti-derivative content? Just I will tell you the first thing is what is this anti- derivative anti-derivative in the sense suppose to if you take an as if you take an as d by dx and this d by dx if you are taken d by dx of f of x is taken okay now so this is you are taken let us see okay so anti option area So if you are apply some other operation it can be removed. It can be removed means differentiation will be vanished.

[00:03:27]Differentiation complete possible only it's anti-derivative.

[00:03:33]Anti-erative is nothing but what?

[00:03:35]Integration. Now the integration integration integration integration integration is is anti-derivative anti-t anti-derivative process anti-derivative derivative derivative process anti-derivative process we can say is that this is anti-derivative of the given derivative.

[00:04:09]So anti-derivative in nothing but what integration is the anti-derivative.

[00:04:15]Let's I have applied the integration here. So this will be cancelled.

[00:04:18]Integration and differentiation will be cancel which becomes dx which becomes only f ofx. So f of x integration will be finishes means we will write the c here. Understand? So that c we will talk about later. First thing is d by let's say d by dx of sin x what is the happening better d by dx of sin x will be differentiation of sin x will be cos x that we know or t we know so now if I apply the integration of cos x cos x integration cos xation Listen carefully.

[00:05:17]theta. So you want to cancel dx dx cancel integration of d of sin x derivative of sin x equal to integration of cos x into dx which is equal to.

[00:05:32]So integration different sin sin x= integration of cos x dx and integration of cos x dx and integration of cos x dx. What you are getting? What you are getting? Sin x sin x plus c plus integration.

[00:05:57]Okay. Right. So right next it can be applicable to anywhere right so we will check down these all things d by dx of x^ x^ n + 1 / n + 1 = x^ this one. We'll check it out. Check it now. dy dx of x n + n + 1. Right? So n +1 into x^ n + 1 -1 divided by x + already x okay I'm applying the integration just I'm applying the integration integration I'm applying here integration I'm applying with respect to dx integration with respect to dx. So integration differentiation cancel dx dx cancel.

[00:07:25]Next integration of d of x^ n + 1 / n + 1. Okay. Now equal to integration x^ n into dx into dx. Right? So d integration cancel integration x^ n into dx= x^ n + 1 / n + 1 integration.

[00:07:56]This is a proof for the differentiation of x^ n + 1 by n + x integration x^ n + 1 by n + 1. Next. Next. Next part.

[00:08:14]dy dx of x d by dx of x. This is equal.

[00:08:22]This is equal to one. Integration applies to both sides. Integration dx.

[00:08:27]Okay. Now, so dx and dx dx cancel. D of X integral = 1 integral DX cancel integration 1 DX= X that is integration 1 DX= X right next part. So differentiation of suppose to integration of log x integration of log x dx it cannot be processed but differentiation of d by dx of log x will be 1x that we notice already.

[00:09:09]So 1x integration integration of 1x will becomes log x but log x integration never be 1x log x integration we check down after it is x log x - x will okay right next part integration of e power x beta into dx which is equal to e power x only integration of a power x.

[00:09:44]Similar a power x into log 1 e power x by 1 + c which is equal to e power x + c that's answer right similar next part next part. Next integration of sin x will becomes how much? It can be min - cos x + c. Integration of cos x dx will be what? It is sin x plus c of beta.

[00:10:24]Integration of but integration of integration of tan x tan x dx will be log of se x. It will be log of se x ala.

[00:10:39]Okay. Integration of c x c x dx which is equal to how much? will be log of cosec x a cosec x plus or minus minus otherwise otherwise log logus cos inverse it will become inverse cosec inverse means sin x plus c both are same both are Cortex integration of se x dx will be equal to log of se x + tan x tan x + c.

[00:11:28]Integration of cosec x dx will be equal to log of cosec x - c x + c.

[00:11:42]Okay. So point of difference.

[00:11:50]So next important discuss.

[00:11:55]So integration of second square x² x dx will be equal to tan x beta plus c.

[00:12:04]Integration of cos x² dx is equal to minus c x. It will be it will be minus c x + c. Next integration of se x and tan x dx is equal to se x + c. Integration of cosec x into c x into dx is equal to minus cose x beta plus c. So next there is a very very important part that is that is a very important part which that is that is just you can check down and see properly you can see that what is that means now integration of 1 by 1 + x² into dx which will becomes which will become terms tan inverse of x + c beta. Okay. Now integration of 1 by root roo<unk> / 1 - x² which into dx which is equal to sin universe of x + c sin inverse of x + c.

[00:13:32]Now integration it will become inverse of x plus cos inverse of x that's it right. So once difference next part so integration of 1 by into <unk>x² -1 into dx which is equal to sec inverse of x cus cose x cos cosec inverse of x plus Both are same.

[00:14:34]Okay. So next hyperbolic functions hyperbolic function differ only positive value of the cos. Next uh sec h² x sec hyperboly function equal to tan hx cosecus sec x tan x and minus x cosec x and minus cosec x cose x right so point of discussion is very very important this is very important integration 1 by<unk> / 1 + x² into dx which is equal to sin S of H inverse of X plus C important 1 + roo<unk> 1 - square okay sin inverse hyperbolic function this is a hyperbolic function which you are getting hyperbolic function so this can be equivalent to the equivalent to log log base e beta log base e. So that is equal to x +<unk> / x² + 1 or 1 + x² it is modulus plus integration completed right next integration of 1 by<unk> / x² - 1 x² - 1 into dx which is equal to cos of h inverse of x + c form 1x roo<unk> 1 + x 1 - x² inverse x cos inverse but hyperbolic function x²us 1 + x² 1 - x² next x² - 1 right which are important this equal log Log important base e log base e same formula x + root roo<unk> x² + 1 + x² 1 x² roo<unk> x² - next back Sorry. So integration of 1x 1 + x² dx will becomes how much? I told you this is this is how much? It will be tan universe of x cas. Okay. So integration of 1 by 1 - x² integration of 1 - x² will becomes tan inverse of x tan h inverse of x hyperbolic function. So integration of 1 by 1 - x² dx will becomes what tan h inverse of x otherwise plus c otherwise or or else 1x 2 * of log 1x2 * of log 1 - x² denominator 1 - x² 1 + x by 1 - x also d + c. This is the part the total explanation part of the trigonometric and different type of the formulas. Next partation rules first integration first.

[00:18:49]So if you want it to be write the integration of the any function the function should be in a additional terminology additionally.

[00:19:08]So f of x plus or minus either plus or minus of g of x f dx g of x dx1 k2 k3 k4 constants integrations separate Integration let it be suppose to one example for this on this basis I'll give you the one example integration over e power e power x + 4 x² + sin x into dxle Okay.

[00:20:07]So, integration of e power x dx plus 4 * of 4 * of x² integral dx plus integration of sin x dx. Right? If I take the integration e power x is okay. Okay.

[00:20:32]Plus 4 constant. Let's take it as outside. x² into dx plus integration of sin x dx. Okay. Sin x dx e power x integration e x. Okay.

[00:20:52]x² as per the x integration x^ 2 + 1 by 2 + 1 x^ 3 similar x^ 3x 3 plus sin x will becomes - cos x integration finish apply the c here So this is a solution for that one.

[00:21:23]Right? One more example. If I try to take one more example. So integration x cq + x² + 4x + 1 by <unk>x is there into dx equation.

[00:21:43]So individual.

[00:21:48]So divide will be applicable individually. So integration x cq /<unk>x plus integration x² /<unk>x plus integration 4x /<unk>x plus integration 1 by x into 1 by<unk> xation DX right next dx plus integration of x² into x^ -1 by 2 into dx / t. Right? Next, integration of 4x into x^ - 1 by 2 into dx / t plus integration of x^ - 1 by 2 dx - 1x 2 dx t. So integration x^ 3 - 1 by 2 dx plus bal.

[00:23:20]So x power 2 - 1 by 2 into dx 4 constant x^ 1 - 1 by 2 plus into dx integration x^ - 1 by 2 into dx. Okay.

[00:23:39]Simplify integration of x 2 into 3 6 - 1 5 by 2 into dx plus integration.

[00:24:05]Next x^ 2 2 2's are 4 - 1 4 - 1 3x 2 into dx plus integration of four constant first x^ 1 - 1 by 2 1 full quantity 1 by halfantity Okay.

[00:24:37]2 into 1 2 2 - 1 1 by 2 1 by 2 into dx + dx right. So x power 5x 2 + 1 divided by 5x 2 + 1 + x power 3x2 + 1 divided by 3x2 + 1 as per the x power n means x^ n + 1 by n + 1 format right + 4 * of x^ 1 by 2 + 1 divided by 1x 2 + 1 + x - 1 by 2 + 1 / - 1 by 2 + 1 + c. This is a existing solution but it was not completed because because next part simplify 1 into 2 2 + 5 7 by 2 7 by 2 becomes 2x 2 x 7 by 2 Next 2 1's are 2 2 + 3 5 by 2 x power 5 by 2 same four *s of x power 2 2's 2 1 are 1 + 1 3x 2 3x 2 + you got 2 - 1 will be 1 by 2 x^ 1 by 2 + 1 plus 2x 7 x^ 7 by 2 + 2x 5 x^ 5x2 + two multiplication by 3 x^ 3x 2 + 2 into x^ 1 by 2 + c beta existing solution which one for which one existing solution okay now so kind of questions.

[00:26:56]Right. Next part. Next part. Suppose to integration first item. Trigonometry.

[00:27:09]Trigonometry.

[00:27:11]Trigonometry formulas. Trigonometry formulas.

[00:27:17]Next important expressions Expressometry functions this part is very easy that is the part right. So discussion is very very clear problems.

[00:27:59]It's all depends upon formulas only formula based example is sin of x plus cos of x plus tan x into dx. What is the integration?

[00:28:17]Simple sin x into dx cos x into dx plus integration integration tan x into dx. Understand?

[00:28:34]So sin x dx will be sin will be minus cos x.

[00:28:39]Integration of cos x will be sin x.

[00:28:43]Integration of tan x will be c x + c.

[00:28:48]Yes, this is correct. No, it's how it will be mean already formulas problem integration of tan x will be how much?

[00:28:58]What I told you? Integration of tan x will be how much?

[00:29:06]Integration of tan x will be how much?

[00:29:09]log of tan will be log of sec x + c.

[00:29:15]This is answer direct formula is direct formula.

[00:29:21]Okay. So next part second problem integration of sin² x + cos² x dx. How much it will be? First solution.

[00:29:38]Integration of sin² x + cos² x into dx is equal to e formula answer formula 1 1 dx 1 dx and x + c. Answer is what is this answer? x + c is the answer.

[00:30:08]formula.

[00:30:12]That is a part. Next part. Next part is integration of sin² x into dx. Okay. sin square x formula cos square x formula integration of cos² x for so integration of sin square x 1 1 - cos 2x by 2 into dx integration 1x 2 dx minus integration of cos cos 2x / 2 into dx right so 1 by 2 is constant integration 1 dx 1 by 2 is constant integration of cos 2x into dx okay so 1x 2 into 1 dx means x - 1x 2 into cos a sin 2x by 2 + c okay cos will be sin 2x. Okay, that is senator.

[00:31:34]Next solution 1 + cos 2x / 2 into dx, right? 1x 2 1 dx plus 1 by 2 into cos 2x into dx right 1 by 2 1 dx and x 1x2 into cos 2x sin 2x / 2 + c this questions is very important questions very important very very important right so point of third question which is very important 1 - cos 2x / 1 + cos 2x into dx beta problem eas 1 + cos cos 2xide with 2x into dx.

[00:32:52]Okay. So sin² x cos² x dx integration tan² x into dx.

[00:33:06]Okay.

[00:33:14]minimum conversion college students. So some of the questions are very very important. So these parts are very very important to the students. So here uh some of the very very important questions which last time it was asked in the examination uh that we will see now. So integration over 1 by <unk>x plus roo<unk>x plus sin 2x sin x. So integration dx. So this guys a question.

[00:33:47]And one more question is uh e powerx - 1x e power x - 1x + 2x<unk> / x² - 1 then + 1 by 1 + x².

[00:34:11]So this is the one more problem into dx.

[00:34:17]So here you can set down this v problem while while you are doing this problem you understand something. So here integration over so 1 by <unk>x dx is the one and integration rootx into dx plus integration sin x dx.

[00:34:39]Okay. So this root x can I represented x power 1 by 2 1x of dx plus integration x^ 1x2 dx plus integration sin x dx integration this will be x^ - 1 by 2 dx plus this will be x^ 1x 2 dx now plus integration of sin sin x into dx.

[00:35:13]Okay, whatever it may be. Suppose to we convert into the integration formats.

[00:35:18]But now we are applying x power it will be - 1 by 2 + 1 / - 1 by 2 + 1 + x power 1 by 2 + 1 / 1 by 2 + 1 + sin x will be cos x which cos x it should be minus cos x integration will be means see you need to apply now x power - 1x 2 + 1 becomes 1 by 2 divided by 1 by 2 plus this is x^ 3x 2 / 3x2 - cos x + c right so next point is very clear so integration you have to convert that is into the formula format integral e power x into dx is there integration 1x into dx is there plus integration 2 is constant 1x<unk> x² - 1 dx + integration 1 by 1 + x² dx e power x this will be log x 1x will be log x now right so 1 by there is a + 2 into + 2 into 1x<unk> / x² - 1 x² - 1 will becomes what?

[00:36:52]So it is already we noticed that uh 1 - x² 1 by<unk> over 1 - x² will be the sin inverse of x if 1 by it will be the roo<unk> / x² - 1 is there. So you can back to the our formula discussion which we done. So here roo<unk> / x² - 1 roo<unk> / x² - 1 means cos h inverse of x the answer is this will be cos h inverse of x plus this is 1x 1 + x² means tan inverse of x + c. So this is the story behind this problem.

[00:37:41]So questions kind of conversion basic formulas similar some of the questions are very easy and you can write these problems.

[00:37:52]You can try these problems. The problem is first problem is here.

[00:37:58]First problem is here that is uh x^ 5 + x^ 5 + 5^x + 5x don't get confused this problem. So integration dx. So here this is x^ 5 integration x^ 5 into dx plus integration 5^x into dx plus integration 5x into dx.

[00:38:31]So it will be suppose if you apply x^ 5 + 1 divided by 5 + 1 it should be 5^ x / log of 5 as per the formula 5 is constant x power will be 1 so 1 + 1 divided 1 + 1 integration is over plus c you have to add it so this becomes x^ 6 / 6 + 5^ x / log 5 + 5 into x² / 2 + a solution.

[00:39:10]So second problem in this second problem is 2 into e power x + 2 into e power x 3 into sin x + 4 sec² x.

[00:39:29]So you need to give the integration for this integration.

[00:39:32]So for integration of this one it is very simple. So 2 is constant 2 e^x dx is the first term. After the plus plus integration 3 into sin x + 4 into once your integration will be over you have to return dx. Okay. Plus integration of sec² x into dx. Right. Two constant integration e^x into dx + 3 into integration sin x into dx + 4 into integration sec² x into dx. Okay.

[00:40:12]Existing formula is what? Beta for here.

[00:40:16]2 * of e power x will be e power x 3 into sin x will be cos x plus 4 into sin x will be minus cos x this is very important right next is 4 into see square x means see square x means it is tan x plus c.

[00:40:35]So that is the question. So this is the procedure you are solving these problems is based on the where so b into integration sin x into dx + 4 into integration see² x into dx. Okay, existing formula is what? Beta for here 2 * of e power x will be e power x 3 into sin x will be cos x plus 4 into sin x will be minus cos x. This is very important right. Next is 4 into sec square x means sec square x means it is tan x plus c.

[00:41:19]So that is a question. So this is the procedure you are solving these problems is based on the where? So based on the based on the this content of the already formulas we written these formulas based on these formulas we written. So this if you understand these formulas properly you can apply this those formulas to the existing problem it will be easy to solvable that is easy to solvable. Okay.

[00:41:44]So some of the questions which are very very important questions I will going to be discuss now it is a relatively trigonometric problems. So when the trigonometry problems is coming so you need to comple compulsory understand the things I told you already this is very very important things integration law once what we doing means integration root over 1 + let's say cos of 2x is there 1 + cos 2x is there so kind of problemsome don't be get confused because important question. So 1 + 2x 1 + 2x formula. So we get a cos 2x formula cos 2x = two formulas cos 2x = 1 - 2 sin² x cos 2x formula that is equal to 1 + cos 2x for cos 2x is another formula cos of 2x cos of 2x equal to 2 cos² x - 1 these are two formulas we have suppose what they asking 1 + cos 2x this one if I take this side in the right right side it will becomes what 1 + cos 2x = 2 cos² x I can say that this can I replace with this what I can root okay 2 * of cos² x into dx now okay here integration <unk>2 is there root cos² x is there into dx is there so roo<unk>3 square cancel so <unk>2 is taken as outside now <unk>2 is taken as outside so it is cos x into dx is there integration of cos x will be what beta Integration of cos x will be sin x integration over means plus c you have to write. So this is based on the performance which one trigonometry equations equations.

[00:44:36]So trigonometry formulas problem solv next next point is next point is integral of next point is integration of roo<unk> over roo<unk> over integration roo<unk> over 1 + 1 + sin 2x 2 theta sin 2 theta So for this kind of question just you can understand it is with respect to dta beta. So 1 + sin 1 + sin 2 theta right.

[00:45:16]So can I represented this is a problem integral over. So this one I can choose as a cos² theta + sin² theta plus sin 2 theta. First one equal to I can say is that sin square theta + cos² theta sin 2 theta can I represented with the 2 * of sin theta and cos theta so which becomes 2 * of sin theta and cos theta right into dta so integral over roo<unk> over so this is a² This is b² plus this is 2 * of a into b.

[00:46:05]I can represent this is equal to a + b whole square. What is the a value? cos theta plus sin theta.

[00:46:15]It's all square root cancel. Integration into d theta is there. Integration will be cos theta plus sin theta into d theta. Integration cos theta into d theta plus integration sin theta into d theta right.

[00:46:41]So cos theta means sin theta plus sin theta means minus cos theta integration plus c. So that is the answer that is answer. So these kind of the questions which is very very important questions in the examination you might get the these kind of the questions for the purpose of the examiner will try to test so next here. So this kind of the question 1 by 1 - sin x sin x either theta into dx so 1us 1 + sin x multiplication and division 1 + sin x into dx integration 1 + sin X divided by 1 square - sin² X into DX integration 1 + sin X / cos² X into DX integration 1 cos² x plus sin x divided by square integration dx right. So sin x into dx right. So integration see² x into dx plus integration is sin by cos will be tan x into sec x into dx right.

[00:48:56]So direct formula second square direct formula second square x direct formula se x square x means tan x plus tan x means se x plus c. So okay.

[00:49:19]So this is one more problem right? So kind of questions number of times integration you can try for this one integration 1 by 1us cos x you can try this problem into dx 1 by 1 - cos x 1 + cos x / 1 + cos x right integration 1 + cos x divided by 1 square - cos x² into dx integration. So this can represent 1 - cos² x means sin² x.

[00:50:04]So 1 + cos x into dx.

[00:50:10]So next individual denominator 1 by sin² x plus cos x divided by sin² x into vxa right integral 1x sin sin square x means it should be 1x sin square x will be becomes what we cose square x. Okay. Uh once you written the integration, you have to written separately dx. That is very important, right? Integration of this is the cos x divided by sin x into 1 by sin x into dx. Right? So now integration cos sec² x dx plus integration over cos x by cos x by sin x will be it will be c x 1x se x means cos x 1x sin x means cosec x cosec x into dx I can say that so this can be represented by the integration of the integration of the integration of the secant square x means integration of the sec square x means minus c x plus c x se x means minus cosec x plus c so that is the integration of this one so these kind of the questions we'll try to push the examination so So right total ability of the student we checking with the trigonometry.

[00:52:10]Okay.

[00:52:14]So right next questions we can solve it. Okay. Okay.

[00:52:24]So what is the kind of equation is sin² x divided by 1 + cos 2x 1 + cos 2x integral dx. This is very simple. So I can say that this will be sin² x. Okay.

[00:52:44]1 + cos 2 already 2 cos² x - 1us 1 + cos 2x = 2 cos² x 2 cos² x into dx 1x 2 into sin square by cos² will tan square x into dx but tan square x² x - 1 into dx right 1 by 2 into sec² x dx - integral of dx 1x common right it is common now 1x2 into 1x2 Seek square x means square x means what is the sec square x integration already what is square x integration of the second square x.

[00:53:58]So check down the direct square x means tan x. Okay. Now see square x means it is a tan x.

[00:54:16]It will be the tan x.

[00:54:20]So this will becomes 1x2 into tan x - x. 1 x means - x plus c. So that's the solution of this one. So particular topic important. So compressor question.

[00:54:38]Okay. So once next integration by substitutions method, integration by substitution method. So integration by substitution.

[00:54:52]ities. Okay. So, integration by substitution method.

[00:54:59]Okay. So integration of integration of f of x into dx = capital f of x + c and so integration of f of x a x integration of f of a x + b into dx a.

[00:55:51]So capital f of a x + b divided by x coefficient a plus c.

[00:56:09]Okay. So this is integration by substitution you have to check this one. So important is taken as so like derative both sides derative a is constant x will be becomes dx b will be become zero equal to dt a into dx is equal to dt dt. So dx becomes 1 by a * of dt.

[00:56:51]So information.

[00:56:54]So integration fn dxal 1x a into dt 1x conant is outside f of t into f of t into dt.

[00:57:12]Now 1 by a into f of t means f of t means capital f of t plus c. Okay. So 1 by a capital f of t a x + b plus c integration by substitution method integrations.

[00:57:38]integrations by substitution method. Substitution method.

[00:57:50]Hello students, welcome to Ask Engineering Academy. Right. So this is the second chapter in the unit number unit number two actually it is comes under the it is common to every C24 regulation students.

[00:58:06]And uh this is the this is a syllabus normally what they try to asking in this area because previous C21 important part of the discussion is sin to the power of nx cos to the power of n mx into dx into model important model. Next model sin a cos bx sin a x sin bx. So models cos of a x then cos bx e mod comes under the same model. This comes under the same model. integration.

[00:59:09]Let's see. Let's see. So 1 by x² + a² into dx² + a² x differentiable integral variable.

[00:59:22]So a constant number right. So integration of 1 by a²us x² so previous now integration of x² - a² into dx pattern pattern same question is suppose to it will have the log so logarithm integral root 1 by roo<unk> / x² + a² into dx.

[01:00:10]So without root the same formula integral of 1 by roo<unk> / a² - x² into dx. Then next integration of 1 by<unk> / x² - a² into dx differences. Okay.

[01:00:37]Similaration x² + a² into dx integration roo<unk> over a² - x² into dx integration root over x² - a² into dx. So patterns integration of 1 by a + b into sin x dx integration 1 by a + b a plus or minus b cos x dx integration 1 by a + or minus b uh sin 2x X cos 2x integration 1 by a + b cos 2x for cosm.

[01:01:44]So e format format let's see see here integral of 1 by a into sin x + b into cos x + c plus or minus plus or minus right. So integration of 1 by a into a into cos x + b sin x plus c for into dx into dx. So format this is a question examiner number it is covered on based on this area totally.

[01:02:29]So let's see first clarity first thing is we need some clarity about the function existing formulas we'll discuss it later first this is supposed to these two values are positive 1x a into tan inverse of x by a + c 1 A intoverse that is the first point. Next point is so 1 by a square - x² x - okay x a square a into log of log of a + x by a - x a + x by a - x + c. So x 1 by 2 a into log of log of x + a by sorry x - a you will get x - a by x + a x - a plus right so which are important right so you come to kind of questions problems So problem integration 1 by x² + 4 into dxic.

[01:04:47]Okay. So now integration 1 by x² + 4 square into dx.

[01:04:56]So 1 by a 1 by a a means here 2 into tan inverse of a x by plus c plus c x2 + c.

[01:05:18]So this is existing solution right?

[01:05:20]supposed to the second problem is on this level only integral of 1 by x² + let's say 7 into dx but previous now we cannot convert into two square means square simple technique. So that is integration 1 by x² + dx integral 1x² +<unk> 7 square into dx<unk> 7 square into dx. Okay.

[01:06:11]So it is perfectly perfectly applicable right. So xic for x² + a square for 1.

[01:06:31]So 1 by<unk> 7 into tan inverse of a x by<unk> 7 plus c. So same format of question. Same format of question. Suppose to integral of 1 by x² + 4x + 5 x² + 4x + 5.

[01:07:04]So a² + 2 * of a into b + b²= a + b square a + b square right. So similar x² + 4 x + 5 a + b square right x² into x + 5 right x² + 2 * X into square.

[01:08:11]Now as per the so 4 + 1 and x² + 2 * of x into 2 + 2 2² + 1 okay can take this total entire term a square format 2 x and 2 and a coefficient x + 2 square + 1 x + 2 square + 1. Now solution is very simple. Once the problem solution is very simple integral 1 by x² + 4x + 5 into dx. So integral of 1 by integral of 1 by so this is given to me right so this is x + uh can I represent this is x + 2² + 1 into dx right so I can say is that 1 by x + 2² + let's say 1 square will be 1 1 square right now this is equal to this is equal 1 by 1 + integral of 1 by x² + a² into dx which is equal to 1 by a tan inverse of x by a format integration right tan inverse of tan inverse of x + 2 x + 2 divided by the a value is 1 plus c and the answer tan inverse of x + 2 / 1 plus did you understand majority important discussion is this is majority important discussion majority discussion part is very very important solution Okay. So% square + 4.

[01:10:57]So first square into 2 into x right 4 + 1 4 + 1 will be five square.

[01:11:27]Okay. Remaining a square + 2 a + b square x². So x² into a into b. So 2 square right. So x² + 2 a 2 x b into 2 a into b square + 1.

[01:11:52]So existing formula so this one more question one more question from this area one more question from this area integral of 1 by 1 by x² - 4x + 9 + 9 - 4x + sign into dx. So don't get panic and confused here. So this is very simplest version. We can answer these simplest problems here also. Right? So first existing a square a formula base suppose to a² - 2 * of a into b + b square should equal to a minus b square right this is a very important part already but we need to convert that existing one into the existing format convert So existing existing x² - 4x + 9 right 2 * 2 into right square 9 + 5 9 4 + 5 right x² - 2 * of 2 into x + 2² + 5 + 5 right so this total part I can make it as a a minus b and x - 2² square + okay only <unk>2 square 11 square 12 11 square root 15 square. So roo<unk>3 square root right square plus<unk> square right solution easiest. So solution 1 by x² - 4x + 9 into dx x - 2² +<unk> 5 square into dx right 1 by<unk> 5 1x roo<unk> 5 existing format x² + a² into dx= 1x a intoverse x for 1 into tan inverse of a tan inverse of x - 2 by<unk> 5 whole + c. So that is a solution of this equation. Okay, this is very easy if it is knows problem same integral of suppose to 1 by 4x² + 5 into dx 4 x² + 5 into dx only this is the concept I told you that it should be coefficient as a one but what is the coefficient is there the coefficient is four here you need to remove that four four coefficient then the problem will be able to solved because existing formula.

[01:16:34]So very simple that is also very simple.

[01:16:36]So just 4x² 4 x² + 5. So this is 1x right x² plus 5x 4 into dx. So divide and multip.

[01:17:03]So existing equation divide with multiplication with four multiplication with four right constant integration participate 1 by so x² + 5x4 beta 5x4 but it is not comes under the existing formula. What is our existing formula? There is a plus is there plus 1 by x² + a² into dx. So this is equal to 1x a into tan inverse of a x by a x + c. But here question x² +<unk> 5x4 square into dx into dx roo<unk> 5x4 applicable to entire right. So 1 a value is roo<unk> 5x4 into tan inverse tan inverse of x by a roo<unk> 5x4 plus c right. So this is existing solution.

[01:18:27]It is time taking process. It will not answer. No problem. Right.

[01:18:41]Integration of 1 by 1x² + 4x.

[01:18:48]Let's say 4x 4x -3 into dx - 13 dx okay so almost I told you this point very clearly very clear integral of 1 by x² + 4x - 13 into dx first a whole square for a square + b 2 * of a into b + b² which is equal to a + b square which is equal to a + b square. Now the question of examiner is x² + 4x -3 x² + 4x - 13 1 minute break this 2 minutes break I will come Okay students right this is a x² + 4x + 4x - 30 that is so I can say that this this is will be represented by the similarly x² plus this is a 2 * of 2 * of four will be represented how 2 2 into 2 then this X is similarly X but here I need actually what here I need actually plus plus B square but there is a minus3 is there we have a minus3 is there that minus 13 we cannot use there is a +4 we need now so +4 we need means +4 we can write and when you are writing the +4ractus adding adding of 13 and subtracting of 13. Right? Similarly, sorry x² + 2 * of x into 2 + 2²us - 4 + 13 + 17 right. So x + 2² - 17 - 17. So square constant getting square + 2 square minus 17 roo<unk> again square. So this is the existing format important part.

[01:26:12]First part important part. First part is a b square for yes.

[01:26:41]So square means equation balance. So four equation is balanced. Equation is balanced. So a + remaining - 13 add so 13 - 4 - 17 remaining a a square + 2 * of a into b + b square for a + b square format right so existing equation will be now converting into the 1 by x + 2² -<unk> 17² into dx. Okay. So format integration of 1x² - a² into dx which is equal to what 1x 2 a into log * of a log * of x - a by x + a x + a plus c. Right? Similar roo<unk> 17 is the a value roo<unk> 17 is the a value then answer 1 by 2 * of<unk> 17 log of log of log of x value x + 2 x + 2 minus a value<unk> 17 divided by x + 2 +<unk> 17 plus C that is the existing solution of this one solution the major key role plays a conversion technique this is the only the major key role plays major key role plays major key role plays so the major key role is playing the major key role plays major key role plays conversion important once conversion Perfect solution.

[01:28:58]Okay. Right.

[01:29:01]Right.

[01:29:10]So then we back to here.

[01:29:13]So x² example integration of 1.

[01:29:35]Okay. 13 + 4x 13 + 4x 13 + 4x - - 4x - 13 - 4x - x² 4x - x² 4x + x² right 13 - 13 right 13 - x² + 2 * of 2 into x 2 + 4 - Yes.

[01:31:06]Right. That is 13 - 13 - x² + 2 * of x into 2 + 4 The total terminology this total terminology I can represent it by the a² + 2 * of a into b + b square for that is equal to a + b square for yes a whole squareus a x + 2² square us 13 - 4 9 + 9 - 9 + 9 - x + 2 square x + 2 square 3 square - x + 2 square right so once conversion problem. So once you need a conversion first conversion problem solution solution so integration of 1 by 3²us x + 2² into dx x + 2 square into dx point is very very important format integral of 1 by the constant square minus variable square into dx the constant minus varable square var a - a + x by a - x plus c a + x by a - x + c. So similar 1 by 2 * of 3 into log * of log * of a means 3 plus x means x + 2 divided by a means 3us x + 2 of minus x + 1 Next step.

[01:33:58]So log of x + 5 x plus c. This the final solution. E type of questions. A type of E type of question. Okay. So it is very good enough problems.

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