Integration of the simplest shots of type 1 and 2. Integration of the simplest (elementary) fractions. The method of adding the sign of the differential for the simplest fractions

Guess what shot-rational name the functions of the form $$ f(x) = \frac(P_n(x))(Q_m(x)), $$ y the double fold %%P_n(x)%% i %%Q_m(x)% %.

If %%m > n \geq 0%%, then the rational difference is called correct, otherwise - wrong. Vikoristovuyuchi the rule of subdivision of polynomials, an incorrect rational fraction can be seen by looking at the sum of the polynomial %%P_(n - m)%% of degree %%n - m%% and the correct fraction, tobto. $$ \frac(P_n(x))(Q_m(x)) = P_(n-m)(x) + \frac(P_l(x))(Q_n(x)), $$ de steps %%l%% rich term %%P_l(x)%% less step %%n%% of rich term %%Q_n(x)%%.

In this way, the non-values ​​of the integral as a rational function can be shown by the sum non-significant integrals in the form of a polynomial and in the form of a regular rational fraction.

Integrals in the form of the simplest rational fractions

Among the correct rational fractions, they see chotiri tipi, yakі bring up the simplest rational fractions:

1. %%\displaystyle \frac(A)(x - a)%%,
2. %%\displaystyle \frac(A)((x - a)^k)%%,
3. %%\displaystyle \frac(Ax + B)(x^2 + px + q)%%,
4. %%\displaystyle \frac(Ax + B)((x^2 + px + q)^k)%%,

de %%k > 1%% — integer %%p^2 - 4q< 0%%, т.е. square alignment do not wash their roots.

Calculation of unimportant integrals in fractions of the first two types

Calculation of unimportant integrals in fractions of the first two types does not solve problems: $$ \begin(array)(ll) \int \frac(A)(x - a) \mathrm(d)x &= A\int \frac(\mathrm (d) (x - a)) (x - a) = A \ ln | x - a | + C, \\ \\ \int \frac(A)((x - a)^k) \mathrm(d)x &= A\int \frac(\mathrm(d)(x - a))(( x - a)^k) = A frac((x-a)^(-k + 1))(-k + 1) + C = \\ &= -\frac(A)((k-1)(x-a ) ^(k-1)) + C. \end(array) $$

Calculation of unimportant integrals in fractions of the third type

A fraction of the third type is reversible, having seen the new square near the banner: $$ \frac(Ax + B)(x^2 + px + q) = \frac(Ax + B)((x + p/2)^2 + q - p^2/4), $$ so yak %% p^2 - 4q< 0%%, то %%q - p^2/4 >0%%, still significant as %%a^2%%. Replacing also %%t = x + p/2, \mathrm(d)t = \mathrm(d)x%%, we can rearrange the standard and write the integral as a fraction of the third type in the form $$ \begin(array)(ll) \ int \frac(Ax + B)(x^2 + px + q) \mathrm(d)x &= \int \frac(Ax + B)((x + p/2)^2 + q - p^2 /4) \mathrm(d)x = \\ &= \int \frac(A(t - p/2) + B)(t^2 + a^2) \mathrm(d)t = \int \frac (At + (B - A p/2))(t^2 + a^2) \mathrm(d)t. \end(array) $$

The rest of the integral, the victorious linearity of the non-insignificant integral, can be represented by the sum of two i in the first of them, we introduce %%t%% pіd differential sign: $$ \begin(array)(ll) \int \frac(At + (B - A p /2))(t^2 + a^2) \mathrm(d)t &= A\int \frac(t \mathrm(d)t)(t^2 + a^2) + \left(B - \frac(pA)(2)\right)\int \frac(\mathrm(d)t)(t^2 + a^2) = \\ &= \frac(A)(2) \int \frac( \mathrm(d)\left(t^2 + a^2\right))(t^2 + a^2) + - \frac(2B - pA)(2)\int \frac(\mathrm(d) t)(t^2 + a^2) = \\ &= \frac(A)(2) \ln \left| t^2 + a^2\right| + \frac(2B - pA)(2a) \text(arctg)\frac(t)(a) + C. \end(array) $$

Turning to the correct variable %%x%%, the result for the fraction of the third type takes $$ \int \frac(Ax + B)(x^2 + px + q) \mathrm(d)x = \frac(A)( 2) \ln \left| x^2 + px + q\right| + \frac(2B - pA)(2a) \text(arctg)\frac(x + p/2)(a) + C, $$ de %%a^2 = q - p^2 / 4 > 0% %.

The calculation of the 4th type integral is foldable, for which course it is not considered.

The value of the undefined integral of the fractional-rational function is reduced to the integration of the simplest fractions. To that it is recommended to get familiar with the division of the theory of division of the fraction in the simplest way.

butt.

Know the non-values ​​of the integral.

Solution.

So, as the step of the numerator of the integrated function is more advanced than the step of the banner, then for the cob we see the whole part, passing the stump of the rich term on the rich term:

Tom, .

The layout of the taken proper rational fraction on the simplest fraction can be seen . Otzhe,

Subtracting the integral is the integral of the simplest fraction of the third type. Jumping a little ahead, it is significant that you can take it with a path leading to the sign of the differential.

so yak , then . Tom

Otzhe,

Now let's move on to a description of the methods of integrating the simplest skin shots from several types.

Integration of the simplest fractions of the first type

For the implementation of this task, the method of non-intermediate integration is ideal:

butt.

Know anonymous primary functions

Solution.

We know the non-values ​​of the integral, the victorious power of the first, the table of the first and the rule of integration.

On the top of the side

Integration of the simplest fractions of another type

To complete this task, the method of non-intermediate integration is also suitable:

butt.

Solution.

On the top of the side

Integration of the simplest fractions of the third type

For the cob, the non-insignificances of the integral look like sumi:

The first integral is taken by the method of adding a sign to the differential:

Tom,

The standard of the taken integral is reversible:

Otzhe,

The formula for integrating the simplest fractions of the third type looks like:

butt.

Find the insignificance of the integral .

Solution.

Vikoristovuemo otrimana formula:

Yakby didn’t have a formula in us, then yakbi was charged with:

On the top of the side

Integration of the simplest fractions of the fourth type

The first croc is the leading sign of the differential:

Another crochet is the knowledge of the integral of the mind . Integrals of this kind are known from various recursive formulas. (Sorry about the integration of different recursive formulas). For our approach, the following recurrent formula is appropriate:

butt.

Find the insignificance of the integral

Solution.

For which type of integral function the substitution method is victorious. We introduce a new change (to marvel at the integration of irrational functions):

After substitution may:

We came to the knowledge of the fourth type fraction integral. Our mind can have a coefficient M=0, p=0, q=1, N=1і n=3. Zastosovuєmo recurrent formula:

After a return replacement, we take the result:

Integration of trigonometric functions

1. Integral mind the conversion of additional trigonometric functions is calculated in the sum for the formulas: For example, 2.Integrals of the mind , de m or n– an unpaired positive number, which are counted under the sign of the differential. For example, 3.Integrals of the mind , de mі n-boys positive numbers, are calculated for additional formulas of a lower level: For example, 4. Integrals de counted by the replacement of the change: abo For example, 5. Integrals of the mind can be reduced to integrals of rational fractions for an additional universal trigonometric substitution todi (because = [following the naming of the numeral and banner on] = ; For example, It should be noted that the choice of the universal substitution is often reduced to bulky substitutions.

§5. Integration of the simplest irrationalities

Let's take a look at the methods of integrating the simplest types of irrationalities. one. Functions of this type are integrated in the same way, like the simplest rational fractions of the 3rd type: the square trinomial has a new square and a new change is introduced. butt. 2. (Under the sign of the integral-rational function of the arguments). Integrals of this type are calculated for additional replacement. Zokrema, in integrals mean . As a pіdіntegrаlnaya funkіtії to avenge the root of different steps: , then mean , de n– the least ardent multiple of numbers m,k. example 1. butt 2. - Wrong rational drib, visible to the whole part: – – – 3.Integrals of the mind calculated for additional trigonometric substitutions:

44

45 Pevny integral

Integral value- Additive monotonous norming of the functional, tasks on multiple pairs, the first component of which is an integrated function or functional, and the other is the area of ​​the multiple task of the function (functional).

Appointment

Let it be assigned to . Rozіb'єmo on the part with a lot of dots. Then it seems that the splitting of the wind was broken. Then we chose the right point , ,

The penny integral of the function on the cross is called the interintegral sums with the right rank of the split to zero, as it is independent of the split and the choice of points, tobto

If a boundary is defined, then the function is called integrated after Riemann.

Appointment

- lower boundary.

- upper boundary.

· - Integral function.

· - Dovzhina private vіdrіzka.

· - Integral sum of the functions on the average distribution.

· - maximum dozhina frequent.

power

While the function is Riemann-integrated on , it is slashed on it.

Geometric zmist

Singing integral as a post area

The sing integral is numerically more square of the figure, surrounded by the abscissa, straight lines and function graphs.

Newton-Leibniz theorem

[ed.]

(redirected from Newton-Leibniz Formula)

Newton - Leibniz formula or main theorem of analysis we give spіvvіdnoshnya mіzh two operations: taking the first integral and counting the primary.

Bringing

Let the integrated function be given to you. Let's start from what is significant, what

so there is no meaning, like a letter (abo) stands under the sign of the sing integral over the wind.

Given a more significant value and a significant new function . Won is assigned to all values, for those who know that if there is an integral on , then there is also an integral of , de . Let's guess what we care about

(1)

We respect that

It is shown that it is uninterrupted on a vіdrіzka. Right, tell me; also

and yakso , then

In such a rank, without interruption, regardless of whether you can, or you can’t expand; it is important that it is integrated on .

The graph is shown on the little one. The area of ​​\u200b\u200bthe changing figure is healthy. Її zbіlshennya dorіvnyuє ploschі figuri , as if from obmezhennosti, obviously, the right to zero when it is not independent, so that it will be a point of continuity, for example, a point.

Now let the function be integrated on , but without interruption at the point. Let's say that today I'll lose my point, even

(2)

Correct, for the specified point

(1) , (3)

They laid it down, and so it’s good, . Dali, with safety in point for the skin, you can say what for.

what to bring, what is the left part of the nervousness about (1) at .

The transition to the border in (3) shows the reason for the similar view at the point and the validity of the equality (2). With whom, it is necessary to know about the law and the lion, I will die.

If the function is uninterrupted on , then on the basis of the brought more, the function is

(4)

May I go, I will. Again, the function is primary for .

This visnovok is sometimes called the theorem about the integral over the changing upper boundary, or the Barrow theorem.

We have brought, that the function is sufficient without interruption for the future, for whom the primary, assigned jealousy (4). CIM brought the primary reason for whether it is a non-stop function.

Now let's have enough primary function on . We know that de - deyka is fast. Vvazhayuchi have tsіy rіvnosti і vrakhovuuchi, scho, otrimaemo.

In such a manner, ale

Neutral integral

[ed.]

Material from Wikipedia - free encyclopedia

Integral value called obnoxious, so that one of the advancing minds is accepted:

· Boundary a or b (otherwise insulting boundary) є unskinned;

· The function f(x) can have one or more points in the middle of the cut.

[ed.] Neural integrals of the first kind

. Todi:

1. Yakscho that integral is called . In what direction called similar.

, or just let's break it down.

Let it be appointed and without interruption on an anonymous view . Todi:

1. Yakscho , then the signification that integral is called unsatisfactory Riemann integral of the first kind. In what direction called similar.

2. There is no ending ( or ), then the integral is called expanding up to , or just let's break it down.

If the function is assigned and without interruption on the whole number line, then you can use the non-consecutive integral of the function with two infinite integration boundaries, which are defined by the formula:

, de s - quite a number.

[ed.] Geometrical pattern of a non-contiguous integral of the first kind

The non-existent integral hangs over the area of ​​the infinitely long curvilinear trapezium.

[ed.] Apply

[ed.] Neural integrals of the second kind

Let it be assigned to, endure endless expansion at the point x = a . Todi:

1. Yakscho , then the signification that integral is called

called rozbіzhnym to , or just let's break it down.

Let it be assigned to, endure innumerable growth at x=b . Todi:

1. Yakscho , then the signification that integral is called inconsistent Riemann integral of another kind. In this way, the integral is called similar.

2. Like abo, then the sign is saved, but called rozbіzhnym to , or just let's break it down.

Since the function knows the difference between the inner points of the hole, then the non-obvious integral of another kind is defined by the formula:

[ed.] Geometric zmist inconsistent integrations II kind

Non-viscous integral hanging over the area of ​​infinitely high curvilinear trapezium

[ed.] butt

[ed.] Okremy vipadok

Let the function be assigned along the entire numerical axis and may expand at the points.

Then you can know the unknown integral

[ed.] Cauchy Criterion

1. Let it be signed on an anonymous sight .

Todi converge

2. Let it be assigned to i .

Todi converge

[ed.] Absolute comfort

Integral called absolutely similar, like converge.
If the integral converges absolutely, then the vin converges.

[ed.] Umovna zbіzhnist

The integral is called mentally similar, like to converge, but to disperse.

48 12. Unexplained integrals.

When looking at the sing integrals, we assumed that the region of integration is surrounded (more specifically, in the case of [ a ,b ]); for the basis of the sing integral, the substitution of the integrand function on [ a ,b ]. Let's name linear integrals, For those who suffer insults and minds (exchange and areas of integration, and integrated functions) hold on; integrals, for which the values ​​are broken (so either the pintegral function is not separated, or the area of ​​integration, or both at the same time) obscure. Whom have they divided mi vivchimo unsatisfactory integrations.

• 12.1. Inconsistent integrations over non-surfaced interspace (inconsistent integrations of the first kind).
• 12.1.1. Determination of a non-consecutive integral over an inexhaustible interval. apply.
• 12.1.2. The Newton-Leibniz formula for a non-contiguous integral.
• 12.1.3. Signs of alignment for unknown functions.
• 12.1.3.1. Porіvnyannia sign.
• 12.1.3.2. Porіvnyannia sign at the boundary form.
• 12.1.4. Absolute zbіzhnіst nevyshnіh іtegraіv on bezkіnechnomu promyzhku.
• 12.1.5. Signs of the life of Abel and Dirichle.
• 12.2. Inconsistent integrations of nonexisting functions (unexplained integrals of a different kind).
• 12.2.1. Designation of a non-wired integral as a non-circumstantial function.
• 12.2.1.1. Particularity in the left direction of the integration.
• 12.2.1.2. Zastosuvannya Newton-Leibniz formulas.
• 12.2.1.3. Features of the right end of the integration.
• 12.2.1.4. Special features of the internal point of integration.
• 12.2.1.5. Dekіlka features for the interim integration.
• 12.2.2. Signs of alignment for unknown functions.
• 12.2.2.1. Porіvnyannia sign.
• 12.2.2.2. Porіvnyannia sign at the boundary form.
• 12.2.3. Absolute is the intellectual feat of the inconspicuous integrations in the form of different functions.
• 12.2.4. Signs of the life of Abel and Dirichle.

12.1. Nevlasn_integrated over non-bridged space

(negligible integrals of the first kind).

12.1.1. Determination of a non-contiguous integral over an inexhaustible gap. Come on function f (x ) is assigned on the first line and integrated according to any other way [ vіd, mayuchi on uvazi at dermal z tsikh vipadkіv іsnuvannya і kіncіvku іdpovіdnih inter. Now the solution of applications looks more simple: .

12.1.3. Pair marks for unknown functions. We have divided the allowance that all pіdintegrаlnі funktsії nevіd'єmnі in all areas of appointment. Dosi mi vychaliali zbіzhnіst іntegrаl, іnоkshlyuyuchi yоgo: іs іsnuє іnє іnє іnє іnє іnє pervіsnoї іn vіdpovіdny pragnennі (аbo), іntegr converge, іnоkshe - diverge. In case of violation of practical tasks, prote, it is important to put the very fact of profitability in front of us, and then we will only calculate the integral (before that, the primary is often not expressed through elementary functions). We formulate and bring a number of theorems, which allow us to establish the flexibility and versatility of non-existent integrals in the form of invisible functions, not counting them.
12.1.3.1. Sign of alignment. Let the functions f (x ) that g (x ) integr

As we can see below, not every elementary function can be integrated, which is manifested in elementary functions. Therefore, it is important to see such classes of functions, the integrals of which can be expressed by elementary functions. The simplest of these classes is the class of rational functions.

Whether a rational function is possible in looking at a rational fraction, i.e., like seeing two polynomials:

Not intermingling the sleepiness of the mirkuvannya, let's assume that the richly articulated can't have a sleepy root.

If the step of the numeral is lower than the step of the banner, then the drіb is called correct, in the other case, the drіb is called incorrect.

If the drib is wrong, then, dividing the number on the banner (as a rule, I’ll give you rich terms), you can give the drib at the sight of the sum of the rich term and the correct fraction:

here is a polynomial, a is a regular drіb.

butt t. Let the wrong rational drib be given

Dividing the numeral into a banner (as a rule, subdividing the rich terms), we take

Since the integration of polynomials does not become difficult, then the main folding in the integration of rational fractions is in the integration of proper rational fractions.

Appointment. Proper rational fractions mind

are called the simplest fractions of I, II, III and IV types.

Integration of the simplest fractions of type I, II and III does not become very difficult, so we will carry out their integration without any additional explanations:

More folding calculations will require the integration of the simplest shots of type IV. Let us be given an integral of this type:

Let's remake:

The first integral is taken by the subset

Another integral is meaningfully written in terms of

for the admission of the root of the banner of the complex, and then, Dali repaired by the coming rank:

Let's rewrite the integral:

Integrating by parts, matimemo

Submitting this virase to jealousy (1), we take

At the right side there is an integral of the same type, which is also the indicator of the step of the standard of the pintegral function one lower; in such a rank, we hung through. Continuing to walk along this very path, let's go to the leading integral.

As I have already indicated, there are no easy formulas for integrating a fraction in the integral calculation. And this is followed by a vague trend: what is the “fancy” drіb, it is more important to know the type of integral. At the link with the cym, it is necessary to go into various tricks, about which I will tell you at once. Prepared readers can immediately speed up zmistom:

• The method of adding the sign of the differential for the simplest fractions

The method of piece conversion of the numeral

butt 1

Before the speech, the analysis of the integral can be broken and replaced by a change, signifying, but the record of the decision was significantly finished.

butt 2

Know the non-values ​​of the integral. Vikonati revision.

This is an example of an independent solution. Please indicate that the change replacement method does not work here.

Wow, it's important! Apply No. 1,2 - typical ones are often used. In addition, similar integrals are often blamed for the development of other integrals, as well as for the integration of irrational functions (roots).

Glancing at the reception of practice and at a glance, as the senior step of the numeral, greater for the senior step of the banner.

butt 3

Know the non-values ​​of the integral. Vikonati revision.

Let's start picking a number.

The algorithm for selecting a number is approximately the following:

1) It’s necessary for me to organize for the number clerk, but there. What work? I put in the bows and multiply by:.

2) Now I’m trying to open these temples, what do we see? . Hmm ... already better, but there are no doubles at the back in the number book. What work? It is necessary to multiply by:

3) I'm opening the arms again: . And the axis and the first success! Need viyshov! Ale, the problem is that the zayviy dodanok showed up. What work? Sob viraz did not change, I need to add to my own design the same:
. Life got easier. Why can't you organize it again at the number book?

4) You can. Let's try: . Opening the arches of another dodanku:
. Vibatchte, ale, in me, the boulder was on the front crochet, and not. What work? It is necessary to multiply another addition by:

5) I’m starting to open the bows of another dodan for re-verification:
. The axis is now normal: removed from the residual construction point 3! Ale again, to the little "ale", the zave of dodanok appeared, now, I am guilty of adding to my viraz:

If everything is spelled correctly, then when all the arcs are opened, we can see the output number of the integrand function. Verify:
Good.

In this manner:

Ready. In the rest of the addendum, I zastosuvav method of introducing the function of the differential.

If we know how to lead a viraz to a sleeping banner, then we will definitely see a pidintegral function in us. A look at the method of laying out for a bag is nothing else, like a return to bring it up to a sleeping banner.

The algorithm for choosing a number dial in similar butts is better to use on black. For deyakyh beginners, we think and think. I’ll guess a record-breaking step, if I’ve made a pidbir for the 11th step, and the layout of the number book took up two rows of Verd.

butt 4

Know the non-values ​​of the integral. Vikonati revision.

This is an example of an independent solution.

The method of adding the sign of the differential for the simplest fractions

Let's move on to this type of shots.
, , , (coefficients and do not equal zero).

Indeed, a couple of deviations with arcsine and arctangent already slipped through the lesson The method of replacing the change in an undefined integral. Virishuyutsya so apply the method of introducing the function under the sign of the differential and further integration for the additional table. Axis typical stocks with a long and high logarithm:

butt 5

butt 6

Here, take up the table of integrations and prostheses, for some formulas yak zdіysnyuєtsya rework. Reveal respect, like and visit you can see squares in these butts. Zokrema, at the butt 6, on the back, it is necessary to give a banner at the sight Let's send a letter to the sign of the differential. And it’s all necessary to work it out in order to speed up the standard tabular formula .

That marvel, try independently virishity butt No. 7,8, it’s more stench to make short:

butt 7

butt 8

Know the inconsistency integral:

To give you the opportunity to vikonate and re-verify these applications, great respect - your skills of differentiation are on high.

The method of seeing the perfect square

Integrals of the mind (Coefficients and do not equal to zero) by the method of seeing a perfect square, which already figured in the lesson Geometric transformation of graphics.

In fact, these integrals were built up to one of four tabular integrals, as we looked at each other. And reach out for the help of the well-known formulas of the fast multiplication:

The formulas zastosovitsya itself in such a direct way, so that the idea of ​​​​the method is based on the fact that the bannerman should piece together virazi abo, and then turn them into abo.

butt 9

Know the inconsistency integral

The simplest butt, for Yaku with dodanku - a single coefficient(and not the number chi minus).

Staring at the banner, here everything on the right is clearly staring. We begin the transformation of the banner:

It is obvious that it is necessary to add 4. I, if the viraz has not changed - tsy four and see:

Now you can fill in the formula:

Since the transformation is over START Bazhano vikonati zvorotny hіd: everything is fine, no pardons.

A finely designed butt, which can be seen, may look something like this:

Ready. Offering "freebies" folding function under the sign of the differential: , in principle, you can bully

butt 10

Know the inconsistency integral:

Tse butt for independent solution, as a guide to the lesson

butt 11

Know the inconsistency integral:

What work, if there is a minus before? In your opinion, it is necessary to blame the minus for the arms and place the additions in the order we need: . Constant("two" in to this particular type) don't sip!

Now at the temples dodaemo alone. Analyzing the viraz, we come to the visnovka, what is needed for the bow - add:

Here we have a formula, zastosovuemo:

START vikonuemo on black translation:
, What and need to be reviewed.

The finished butt looks something like this:

Simplify the task

butt 12

Know the inconsistency integral:

Here, with the dodanka, it’s not a single coefficient, but a “five”.

(1) If there is a constant, then її it’s immediately culpable for the arms.

(2) First of all, it’s better to blame the constant for the interintegral, so that you don’t respect it under your feet.

(3) It is obvious that everything can be reduced to a formula. You need to go to the dodanka, and yourself, take the “two”

(4) Yep, . So, it’s up to the virazu, and the same drib is visible.

(5) Now you can see the next square. A savage type also needs to be virahuvati, but here we have a formula for the old logarithm. , and I can’t see the sense, why - you’ll become clear a troch lower.

(6) Vlasne, you can fill in the formula , Only substitute for "iks" with us, which does not reflect the validity of the tabular integral. Strictly seeming, one short is missing - before integrating the function, follow the differential sign: Ale, as I already repeatedly bald, tsim often nehtuyut.

(7) At the root of the bagan, open all the arches back:

Important? It's more convenient in the integral calculation. If you want to apply, what you are looking at, not so foldable, skilki need a good technique to calculate.

butt 13

Know the inconsistency integral:

This is an example of an independent solution. Vidpovіd naprikintsi lesson.

Іsnuyu іntegrаli z znamennikam znamennik, yakі zapomogoyu zameni zvoditsya іntegrіlі v zglyadnogo type, about them you can read statі Folding integrals Ale won razrakhovana on the arc of the trained students.

Announcement of the numeral of the sign of the differential

This is the final part of the lesson, the integrations of this type are often completed often! How much has accumulated in the volume, maybe, it’s more beautiful to read tomorrow? ;)

The integrals, as far as we can see, are similar to the integrals of the front paragraph, you can see the stink: otherwise (Coefficients, i do not equal to zero).

That's why we have a linear function for the numerator. How virishuvati so integrated?

Introduce visnovok formulas for calculating integrals in the simplest, elementary, fractions of some types. More folding integrals in fractions of the fourth type are calculated for the additional formula of the reduction. The butt of integrating shot of the fourth type is examined.

Zmist

Div. also: Table of unimportant integrals
Methods for calculating insignificant integrals

As it turns out, whether a rational function like a real change x can be decomposed into a rich term and the simplest, elementary fractions. Є chotiri types of the simplest fractions:
1) ;
2) ;
3) ;
4) .
Here a, A, B, b, c - dіysnі numbers. Rivnyanya x 2+bx+c=0 I don't have real roots.

Integration of fractions of the first two types

The integration of the first two fractions is followed by additional advanced formulas from the tables of integrations:
,
, n ≠ - 1 .

1. Integration of a fraction of the first type

The fraction of the first type substitution t \u003d x - a is reduced to a tabular integral:
.

2. Integrating a fraction of another type

A fraction of another type is reduced to a tabular integral tієyu w substitution t \u003d x - a:

.

3. Integrating a fraction of the third type

Let's look at the integral of the fraction of the third type:
.
Counting yoga in two priyomi.

3.1. Krok 1

We can see in the number book the shot will go like a banner. Significantly: u = x 2+bx+c. Differentially: u′ = 2 x + b. Todi
;
.
ale
.
We omitted the sign of the module, oskilki.

Todi:
,
de
.

3.2. Krok 2. Calculating the integral z A \u003d 0, B \u003d 1

Now we can calculate the integral that is missing:
.

We direct the banner of the fraction to the sum of the squares:
,
de.
We care that you are equal x 2+bx+c=0 don't have a root. Tom.

Zrobimo substitution
,
.
.

Otzhe,
.

Tim himself knew the integral of the fraction of the third type:

,
de.

4. Integration of a fraction of the fourth type

And now, let's look at the integral of the fraction of the fourth type:
.
Calculate yoga u three priyomi.

4.1) We can see in the numeral book the banner of the banner:
.

4.2) Calculating the integral
.

4.3) Calculable integrals
,
vicorist formula given:
.

4.1. Krok 1

We see in the numeral book the bannerman is gone, as we were robbed in. Significantly u = x 2+bx+c. Differentially: u′ = 2 x + b. Todi
.

.
ale
.

Remaining maєmo:
.

4.2. Krok 2. Calculation of the integral with n = 1

Calculable integral
.
Yogo calculation was posted at .

4.3. Krok 3. Visnovok guidance formula

Now let's look at the integral
.

Inducing a square trinomial to the sum of squares:
.
Here.
We are working on a substitution.
.
.

Vikonuєmo transformation and integruєmo vrozdrib.




.

Multiply by 2(n - 1):
.
Rotate to x and I n .
,
;
;
.

Also, for In we took away the reduction formula:
.
Consistently blocking the formula 1 .

butt

Calculate the integral

1. We see a bannerman in the numeral book.
;
;


.
Here
.

2. Calculating the integral in the form of the simplest fraction.

.

3. Zastosovuєmo formula given:

for the integral.
Our vipadka b = 1 , c = 1 , 4 c - b 2 = 3. We write down the formula for n = 2 and n = 3 :
;
.
Zvіdsi

.

Remaining maєmo:

.
We know the coefficient at .
.

Div. also: