Differentiation and integration are equations containing derivatives. The latter, if we adhere to mathematical properties, are divided into ordinary and particular. Derivatives represent the rate of change, and the differential equation describes the relationship between the quantity, which is constantly changing in the process of solving, forming new variables.
A university professor can easily navigate complex operations with integrals, transform them into a single whole, and then prove the calculus by the inverse method. However, the ability to quickly recall the details of complex formulas is not available to everyone, therefore it is recommended to refresh the memory or discover new material.
Value and main application
In the scientific literature, a derivative is defined as the speed that is subject to a function transformation based on one of its variables. Differentiation is the essence of calculus, which can be compared with the beginning of the search for a tangent to a point. As you know, the latter has various forms and requires computational formulas to search. Suppose you need to find the slope of the tangent to the graph at point P. How to do this? It is enough to draw an arched strip through the marked object and raise it up until we get a split line.
A function f in x is called differentiable at x = a if the derivative f '(a) exists on each notation of its domain. Let's demonstrate an example:
f '() = lim (h=0) × f( + h) – f()/h
In order to subject the equation to differentiation and integration of functions so that its location becomes possible at any point x, it should not be interrupted. Having built a schematic image in advance, you can verify the accuracy of the statement. It is for this reason that the region f '(x) is determined by the existence of its limits.
Suppose that y = f (x) is a function of x, then the derivative of f (x) is given as dy / dx. It is also defined as a linear equation, where it is necessary to find the necessary data on y.
However, if we are looking for the derivative of y in the first case, then in the next we have to find f (x) of x.
d/dx × (f(x)) l a df/dx l a
Therefore, the designation of the rate of change of the function f (x) with respect to x at a point lying on its surface.
If the derivative f 'is known, which is differentiable in its domain, then we can find its value f. In integral calculus, we call f the antiderivative or primitive of the function f '. The method of calculating it is known as anti-differentiation or integration.
Types and forms
An equation with one or more members, which includes derivatives of a dependent variable with respect to an independent, is known as a differential. In other words, it consists of a set of numerical values, ordinary or private, subject to changes in the process of solving.
Currently, the following types of differential equations exist.
Ordinary. Simple equality directly dependent on a variable:
dy / dx + 5x = 5y
With partial derivatives:
dy/dx + dy/dt = x 3 -t 3
d 2 y/dx 2 – c 2 × d 2 y/dt 2
Senior ratio. This type is characterized by participation in the order of the differential equation, as shown in the example below, where it is 3. The number is considered the highest of those present:
d 3 y/dx 2 + 5 × dy/dx + y = √x
Functions can have several types, however, it is preferable to use a single quote with characteristic integration and differentiation formulas.
y' = dy/dx
y'' = d 2 y/dx 2
y''' = d 3 y/dx 3
Linear The variable that appears in the equation is raised to the power of unity. The graph of this kind of function is usually a straight line. For example, (3x + 5), but (x 3 + 4x 2 ) does not belong to this type, since it requires a different solution.
dy/dx + xy = 5x
Non-linear. Any integration and differentiation of series with dual methods of obtaining equality - belong to the form under consideration:
d 2 y/dx 2 - ln y = 10
Quick Methods
It is not enough to consider the form to figure out how to cope and put into practice the knowledge gained. Currently, there are several ways to solve a differential equation.
It:
- Separation of a variable. It is executed when the example can be represented as dy / dx = f (y) g (x). The peculiarity lies in the fact that f and g are functions belonging to their values. Due to this, the problem should be transformed: 1 / f (y) dy = g (x) dx. And only after proceeding to the next paragraph.
- Integrating factor method. Used when the example has the form dy / dx + p (x) y = q (x), where p and q are functions of only x.
First-order differential calculations look like y '+ P (x) y = Q (x), because they contain the necessary functions and the derivative of y. A subsequent increase in the name acts on the same principle. For example, derivatives of an unknown function may turn out to be either private or ordinary.
Indefinite integrals
If you are given the speed of your bike when you went for a walk, depending on the time - can you calculate the distance traveled using the data on the minutes spent? This task looks like an overwhelming burden, but the integrals will help to cope with these properties as efficiently as possible, getting the result.
Scientific literature focuses on the fact that they are the flip side of differentiation. Indeed, integration is a method of adding things up. He connects the particles together, creating something new - the whole. The main thing in any similar example: find the indefinite integrals and check the results of integration by differentiation. This will help to avoid unnecessary mistakes.
If you are going to look for the area of any random curve, for example, y = f (x), then use the method under consideration. Remember that only mindfulness will save you from error.
Formulas for the solution
So, having become acquainted with the basic concept of differentiation and integration - inverse calculation through functions, it is necessary to briefly consider some of the basics. They are listed below.
Basic calculation rules
Integrated functions such as f (x) can easily be converted into equality if we imagine an equation like: ∫ f(x) dx = F(x) + C
Here F (x) is called antiproductive or primitive. f (x) is the integrand. dx - acts as an additional numerical agent. C is an integrated or arbitrary constant. x - appears depending on the side of equality.
From the above statement, we can conclude that the integration and differentiation of the series are two processes that are opposite from each other. Together, they appear as one of the types of operations aimed at obtaining the final result performed on the equation itself.
Now that we know more about the features of calculus, it is recommended to highlight the primary differences needed for further understanding:
- Differentiation and integration can simultaneously satisfy the rules of linearity.
- Operations are aimed at finding the most accurate solution, however, imply restrictions on their determination.
- When differentiating a polynomial example, the result is 1 less than the degree of the function, while in the case of integration, the result is converted to another, acting in the opposite way.
- Two types of solutions, as mentioned earlier, are opposite to each other. They are calculated by the integration and differentiation formulas.
- The derivative of any function is unique, but, on the other hand, two integrals, in one example, can differ by a constant. This rule is the main difficulty during the execution of tasks.
- When dealing with derivatives, we can consider derivatives at a point. Almost like in integrals, they provide interval functions.
- Geometrically, the derivative describes the rate of change of a quantity relative to another, while the indefinite integral represents a curve. It is arranged in a parallel direction, and also has tangents during the intersection of uneven lines with others that are orthogonal to the axis representing the variable.
Addition methods
If you are faced with the problem of how summation is used for mathematical operations of integration differentiation, you should carefully familiarize yourself with the basic formulas. They are an axiom in learning, therefore they are used everywhere. Please note that during application on your own examples, the formulas are true only if they start with i = 1.
Partial Solution
Sometimes a function requires a non-standard approach in order to reach the final result and satisfy the conditions of equality. The term-wise integration and differentiation of the series is based on the identity, which is expressed: ∫ f(x) g'(x) dx = f (x) g(x) - ∫ f'(x) g(x) dx
The algorithm of the methodology under consideration is as follows:
- Express an integrated function as the product of two expressions. We denote one of them by f (x) and the other by g ′ (x).
- Now proceed to identify the other two formulas that can be applied when performing the first paragraph. The row will change. By differentiation, we transform f ′ (x) to obtain the expressions f (x). We proceed to the other part - g (x) integrates into g ′ (x). In this case, dx remains in its original form and is not used.
- Insert the resulting expressions into the formula in parts. This ends the procedure, and now you can try to evaluate the new integral on the right, since it has become much easier to understand.
Previously, this method involved integrating in parts using a matrix. The method was successful, but it took a lot of time, because at present it is used less often, in special cases, when the solution is almost impossible to find. To do this, just put f and g ′ in the first row and calculate f ′ and g in the second.
Why integration by parts?
Situations happen different. Sometimes decisions turn out to be much more complicated than at first glance. Therefore, one should single out the main problems that are often encountered in term-wise integration and differentiation of power series. Let's consider two basic rules.
First, the part that we intend to integrate, that is, the one chosen for g ′ (x), we must be able to transform. It is important to do this as quickly as possible. The fact is that complex integration for g rarely leads to an improved integral, increasing complexity. All this negatively affects the freedom of our actions during decisions, and also depends on the degrees, sines and cosines. Let the search for the right answer take time, but lead to the right rather than confusing.
Secondly, everything else, that is, the part that we intend to differentiate and denote by F, should stand out noticeably after the transformation. After a simple procedure, we notice that the new integral will be more simplified than its predecessor.
So, when we combine the two rules and use it when solving, we get the opportunity to take advantage of the differentiation and integration of power functions, which makes sense to be considered in parts.
There is also a way to remove x, which allows you to effectively use transformations in various situations. For example, we can easily integrate by multiplying the function by a polynomial, which we reduce by differentiation.
∫ x 2 sin(3x) dx
∫ x 7 cos(x) dx
∫x 4 e 4x dx
As f, we take the degree of x (in the more general case, the polynomial), and also use g '. Obviously, each differentiation reduces the degree of the number by one, therefore, if in the example it is high enough, apply termwise integration several times. This will help reduce time.
The complexity of some equations
In this case, we are talking about the differentiation and integration of power series. The function can be considered as if x - is a region of the interval of convergence of points. True, the method is not suitable for everyone. The fact is that any functions can be expressed in the form of power series, transforming into a linear structure and vice versa.
For example, given e x . We can express it as an equation, which is really just an infinite polynomial. The power series is easy to notice by calculating, but it is not always effective.
Defined integral as the limit of the sum
Look at the following graphical integration and differentiation.
In order to easily understand a complex function, it is enough to carefully understand it. Let us estimate the PRSQP region between the curve y = f (x), the x axis, and the coordinates “x = a” and “x = b”. Now divide the interval [a, b] into 'n' equal sub-intervals, denoted as follows: [x 0 , x 1 ], [x 1 , x 2 ], [x 2 , x 3 ]…. [x n - 1 , x n ]. [x 0 , x 1 ], [x 1 , x 2 ], [x 2 , x 3 ]…. [x n - 1 , x n ].
x 0 = a, x 1 = a + h, x 2 = a + 2h, x 3 = a + 3h… .. x r = a + rh x n = b = a + nh n = (b - a) / h. (1). , n → ∞ h → 0.
The considered space PRSQP is the sum of all "n" subdomains, where each is defined on a certain mediocrity [x r-1 , x r ], r = 1, 2, 3 ... n. With the right approach, these functions can be differentiated and integrated for a quick solution.
Now look at the ABDM in the figure. Based on it, it is advisable to make the following observation about the areas: (ABLC) <(ABDCA) <(ABDM).
We also note that as h → 0 or x r - x r-1 → 0, all three regions become practically equal to each other. Therefore, we have:
s n = h [f(x 0 ) + f(x 1 ) + f(x 2 ) + …. f(x n – 1 )] = h r=0 ∑ n–1 f(x r ) (2)
S n = h [f(x 1 ) + f(x 2 ) + f(x 3 ) + …. f(x n )] = h r =1 ∑ n f(x r ) (3)
In this case, s n and S n denote the sum of the areas of all the lower and upper rectangles raised above the intervals [x r – 1 , x r ] for r = 1, 2, 3, ..., n, respectively. To represent this in perspective, equation (1) can be rewritten in the form:
s n < (PRSQP) < S n … (4)
In addition, it is assumed that the limit values (2) and (3) are the same in both cases, and only the area under the curve is common. As a result, we have:
lim n → ∞ S n = lim n → ∞ s n = PRSQP = ∫ a b f(x) dx … (5)
Area is also the limit of the space between the rectangles below the curve and above the curve. For convenience, you should pay attention to the height of the figure, equal to the curve on the left edge of each sub-interval. Therefore, the equation is rewritten in the final version:
∫ a b f(x) dx = lim n → ∞ h [f(a) + f(a + h) + …. + f(a + {n – 1}h)]
∫ a b f(x) dx = (b – a) lim n → ∞ (1/n) [f(a) + f(a + h) + …. + f(a + {n – 1}h)]
Conclusion
Differentiation and integration differs among themselves in a number of properties, formulas and opposite changes. One cannot be transformed into another without help. If differentiation helps to find a derivative, then integration performs a completely different action. She adds some parts, is able to help with degrees, reducing them or perfecting an example by simplifying.
It is also used to test differentiated equations. In other words, they act as a whole, which cannot coexist separately, since they complement each other. Applying the rules, knowing many methods, now you are guaranteed to solve complex problems.