One of the fundamental branches of mathematical analysis is the integral calculus.It covers the wide field of objects, where the first - it is an indefinite integral.Position it as the key is that back in high school reveals an increasing number of prospects and opportunities, which describes the higher mathematics.
appearance of
At first glance, it seems utterly integral to modern, topical, but in practice it turns out that he had appeared in 1800 BC.Homeland is officially considered Egypt as have not survived the earlier evidence of its existence.It due to lack of information, all the while positioned simply as a phenomenon.It once again confirms the level of scientific development of the peoples of those times.Finally it was found writings of the ancient Greek mathematicians, dating from the 4th century BC.They describe the method used where the indefinite integral, the essence of which was to find the volume or area of the curved shape (three-dimensional and two-dimensional plane, respectively).The principle of calculation based on the division of the original figure infinitesimal components, provided that the volume (area) of already known.Over time, the method has grown, Archimedes used it to find the area of the parabola.Similar calculations at the same time, and conduct exercises in ancient China, where they were completely independent from the Greek fellow science.
Development
next breakthrough in the XI century BC has become the work of the Arab scientist "wagon" Abu Ali al-Basri, who pushed the boundaries of the already known, are derived from the integral formula for calculating the sums of the amounts and degrees from the first toFourth, using for this we know the method of mathematical induction.
minds of today admire how the ancient Egyptians created the amazing monuments without any special tools, with the possible exception of his hands, but did not the power of the mind scientists of the time no less a miracle?Compared to the present time of life seems almost primitive, but the decision of indefinite integrals deduced everywhere and used in practice for further development.
next step occurred in the XVI century, when Italian mathematician brought Cavalieri method of indivisibles, which picked up Pierre de Fermat.These two personality laid the foundation for the modern integral calculus, which is known at the moment.They tied the concepts of differentiation and integration, which were previously perceived as autonomous units.By and large, the mathematics of that time has been shattered, the conclusions of the particles exist by themselves, with limited scope.Way of association and the search of common ground was the only true at the moment, thanks to him, the modern mathematical analysis had the opportunity to grow and develop.
With the passage of time changes everything, and the notation of the integral as well.By and large, scientists have designated it in his own way, for example, Newton used a square icon, which put an integrable function, or simply put together.This disparity lasted until the XVII century when a landmark for the whole theory of mathematical analysis scientist Gottfried Leibniz introduced as a symbol familiar to us.The elongated "S" is actually based on that letter of the alphabet, as represents the sum of primitives.The name of the integral was due to Jacob Bernoulli, after 15 years.
formal definition of indefinite integral depends on the definition of the primitive, so we consider it in the first place.
The primitive - it is the inverse function of the derivative, in practice it is called primitive.In other words: primitive function of d - is a function D, the derivative is equal to v & lt; = & gt;V '= v.Search the primitive is, the computation of the indefinite integral, and the process is called integration.
Example:
function s (y) = y3, and its primitive S (y) = (y4 / 4).
set of all primitives of the function - this is an indefinite integral, it is indicated as follows: ∫v (x) dx.
Because the V (x) - These are some of the original primitive function, we have an expression: ∫v (x) dx = V (x) + C, where C - constant.Under the arbitrary constant means any constant, since its derivative is zero.
Properties
properties that have an indefinite integral, based on the definitions and properties of derivatives.
Consider key points:
- integral derivative of the primitive is itself primitive, plus an arbitrary constant C & lt; = & gt;∫V '(x) dx = V (x) + C;
- derivative of the integral of the function is the original function & lt; = & gt;(∫v (x) dx) '= v (x);
- constant is removed from the integral sign & lt; = & gt;∫kv (x) dx = k∫v (x) dx, where k - is arbitrary;
- integral, which is taken from the sum of identically equal to the sum of integrals of & lt; = & gt;∫ (v (y) + w (y)) dy = ∫v (y) dy + ∫w (y) dy.
The last two properties can be concluded that the indefinite integral is linear.Due to this, we have: ∫ (kv (y) dy + ∫ lw (y)) dy = k∫v (y) dy + l∫w (y) dy.
To consolidate consider examples of solutions indefinite integrals.
necessary to find the integral ∫ (3sinx + 4cosx) dx:
- ∫ (3sinx + 4cosx) dx = ∫3sinxdx + ∫4cosxdx = 3∫sinxdx + 4∫cosxdx = 3 (-cosx) + 4sinx + C = 4sinx -3cosx + C.
From the example we can conclude that you do not know how to deal with indefinite integrals?Just find all the primitives!But the search for the principles discussed below.
methods and examples
In order to solve the integral, you can resort to the following methods:
- table ready to use;
- integrate by parts;
- integrated by replacing the variable;
- settlement under the sign of the differential.
tables
easiest and pleasant way.At the moment, the mathematical analysis can boast quite extensive tables, which spelled out the basic formulas of indefinite integrals.In other words, there are patterns derived to you and you can only take advantage of them.Here is a list of basic table positions, which can display nearly every instance, having a solution:
- ∫0dy = C, where C - constant;
- ∫dy = y + C, where C - constant;
- ∫yndy = (yn + 1) / (n + 1) + C, where C - a constant, and n - is different from the number of units;
- ∫ (1 / y) dy = ln | y | + C, where C - constant;
- ∫eydy = ey + C, where C - constant;
- ∫kydy = (ky / ln k) + C, where C - constant;
- ∫cosydy = siny + C, where C - constant;
- ∫sinydy = -cosy + C, where C - constant;
- ∫dy / cos2y = tgy + C, where C - constant;
- ∫dy / sin2y = -ctgy + C, where C - constant;
- ∫dy / (1 + y2) = arctgy + C, where C - constant;
- ∫chydy = shy + C, where C - constant;
- ∫shydy = chy + C, where C - constant.
If you want to make a couple of steps lead integrand to a tabular view and enjoy the victory.Example: ∫cos (5x -2) dx = 1 / 5∫cos (5x - 2) d (5x - 2) = 1/5 x sin (5x - 2) + C.
According to the decision it is clear that for the tableExample integrand lacks multiplier 5. We add it in parallel with this multiply by 1/5 to general expression did not change.
Integration by Parts
Consider two functions - z (y) and x (y).They must be continuously differentiable on its domain.As one of the properties of differentiation have: d (xz) + = xdz zdx.Integrating both sides, we get: ∫d (xz) = ∫ (xdz + zdx) = & gt;zx = ∫zdx + ∫xdz.
Rewriting the resulting equation, we get a formula that describes the method of integration by parts: ∫zdx = zx - ∫xdz.
Why is it necessary?The fact that few examples can simplify, relatively speaking, to reduce ∫zdx ∫xdz, if the latter is close to a tabular form.Also, this formula can be used more than once, for optimum results.
How to solve indefinite integrals this way:
- necessary to calculate ∫ (s + 1) e2sds
∫ (x + 1) e2sds = {z = s + 1, dz = ds, y = 1 / 2e2s, dy= e2xds} = ((s + 1) e2s) / 2-1 / 2∫e2sdx = ((s + 1) e2s) / 2-e2s / 4 + C;
- must calculate ∫lnsds
∫lnsds = {z = lns, dz = ds / s, y = s, dy = ds} = slns - ∫s x ds / s = slns - ∫ds = slns -s+ C = s (lns-1) + C.
Replacement variable
This principle decision of indefinite integrals in demand no less than the previous two, though complicated.The method is as follows: Let V (x) - the integral of some function v (x).In the event that in itself integral in catches slozhnosochinenny example, is likely to get confused and go to the wrong solutions.To avoid this practiced transition from variable x to z, in which a general expression visually simplified while maintaining z depending on x.
In mathematical language is as follows: ∫v (x) dx = ∫v (y (z)) y '(z) dz = V (z) = V (y-1 (x)), where x =y (z) - substitution.And, of course, the inverse function z = y-1 (x) fully describes the relationship and the relationship between variables.Important - differential dx necessarily replaced with the new differential dz, since the change of variable in the indefinite integral involves replacing it everywhere, not just in the integrand.
Example:
- need to find ∫ (s + 1) / (s2 + 2s - 5) ds
apply the substitution z = (s + 1) / (s2 + 2s-5).Then 2sds = dz = 2 + 2 (s + 1) ds & lt; = & gt;(s + 1) ds = dz / 2.As a result, the following expression, which is very easy to calculate:
∫ (s + 1) / (s2 + 2s-5) ds = ∫ (dz / 2) / z = 1 / 2ln | z | + C = 1 / 2ln| s2 + 2s-5 | + C;
- need to find integral ∫2sesdx
To address rewrite the expression in the following form:
∫2sesds = ∫ (2e) sds.
denote a = 2e (replacing the argument this step is not, it is still s), give our seemingly complex, integral to basic tabular form:
∫ (2e) sds = ∫asds = as / lna+ C = (2e) s / ln (2e) + C = 2ses / ln (2 + lne) + C = 2ses / (ln2 + 1) + C.
Wrap under the sign of the differential
By and large, this methodindefinite integrals - twin brother of the principle of the change of variable, but there are differences in the process of registration.Consider detail.
If ∫v (x) dx = V (x) + C and y = z (x), then ∫v (y) dy = V (y) + C.
We should not forget the trivial integral transformations, amongwhere:
- dx = d (x + a), and wherein - each constant;
- dx = (1 / a) d (ax + b), where a - constant again, but not zero;
- xdx = 1 / 2d (x2 + b);
- sinxdx = -d (cosx);
- cosxdx = d (sinx).
If we consider the general case when we calculate the indefinite integral, examples can be brought under the general formula w '(x) dx = dw (x).
Examples:
- need to find ∫ (2s + 3) 2ds, ds = 1 / 2d (2s + 3)
∫ (2s + 3) 2ds = 1 / 2∫ (2s + 3) 2d (2s+ 3) = (1/2) x ((2s + 3) 2) / 3 + C = (1/6) x (2s + 3) 2 + C;
∫tgsds = ∫sins / cossds = ∫d (coss) / coss = -ln | coss | + C.
Online help
In some cases, the fault which may be or laziness, or an urgent need, you can useOnline tips, or rather, to use a calculator indefinite integrals.Despite the apparent complexity and controversial nature of the integrals, their decision is subject to a certain algorithm, which is built on the principle of "if you do not ... then ...".
course, very intricate examples of this calculator will not master, as there are cases in which a decision has to find an artificially "forced" by introducing certain elements in the process, because the result is not obvious ways to achieve.Despite the controversial nature of this statement, it is true, as the mathematics, in principle, an abstract science, and its primary objective considers the need to expand the boundaries of possibilities.Indeed, for a smooth run-in the theories is very difficult to move up and evolve, so do not assume that examples of the solution of indefinite integrals, which gave us - this is the height of options.But back to the technical side of things.At least to check the calculations, you can use the service in which it was spelled out to us.If there is a need for automatic calculation of complex expressions, then they do not have to resort to a more serious software.It is necessary to pay attention primarily on the environment MatLab.
Application
decision indefinite integrals at first glance seems completely divorced from reality, because it is difficult to see the obvious use of the plane.Indeed, their use anywhere directly impossible, however, they are considered necessary intermediate element in the process of withdrawal of solutions used in practice.So, back to the integration of differentiation, thereby actively participating in the process of solving equations.
In turn, these equations have a direct impact on the decision of a mechanical problem, the calculation of trajectories and thermal conductivity - in short, everything that constitutes the present and shaping the future.The indefinite integral, examples of which we have considered above, merely trivial at first glance, as a base to carry out more and more new discoveries.
