Integration by parts
The partial integration, also called product integration in the integral calculus is a way to calculate definite integrals and for the determination of root functions. It can be seen as analogous to the product rule of differential calculus. The Gaussian integral theorem of vector analysis with some of its special cases is a generalization of the integration by parts for functions of several variables.
Rule of integration by parts
Is an interval and are two continuously differentiable functions, then applies
This rule is called partial integration. It has your name preserved because in its application only a part of the integral is determined on the left side of the equal sign, namely, and the second term, namely, still contains an integral. This rule is therefore useful to apply when the primitive function to known or can be easily calculated, and if the integral term is easier to calculate on the right side.
Example
As an example, the integral
Considered, the natural logarithm. Putting and, we obtain
This is then
More examples can be found Indefinite integrals and partial integration section of this article. In contrast to this example, only indefinite integrals there are calculated. That is, the integrals are no limits, which, as shown here to happen in the example, can be used in the last step in the operation.
History
A geometric form of the rule of integration by parts can already be found in Blaise Pascal's work Traité des Trilignes Rectangles et de leurs Onglets (Treatise on curve triangles and their ' adjoint body '), which appeared in 1658 as part of the Lettre de A. Benedetto Ville à M. Carcavy. There was not coined at that time, the integral term, this rule was not described by integrals, but by summation of infinitesimals.
Gottfried Wilhelm Leibniz, who is considered along with Isaac Newton as the inventor of the differential and integral calculus, which proved denominated in modern notation statement
It is a special case of control for the partial integration. Leibniz called this rule Transmutationstheorem and shared them with Newton in his letter, which he sent in response to the epistola prior, the first letter of Newton, to England. Using this theorem examined Leibniz the area of a circle and could be the formula
Prove. It is now called the Leibniz series.
Indefinite integrals and partial integration
The partial integration can also be used to to calculate indefinite integrals - ie to determine primitives. For this purpose, usually for the partial integration of the integral boundaries are deleted, so the constant of integration must now be considered.
Rule
Let and be two continuously differentiable functions and the antiderivative of is known, then the rule for integration by parts
A primitive function to be found.
Examples
In this section it is shown in two examples, as with the aid of partial integration of a primitive function is determined. In the first example, no master function is determined. This example shows the need to focus also on the integration constant in determining a primitive function with the partial integration. The second example is the antiderivative of the logarithm and in the third example, a primitive function is determined to a fractional rational function.
Reciprocal function
In this example, the indefinite integral is integrated, and of partially viewed. While this helps not to determine the antiderivative of concrete. However, it illustrates that must be paid to the integration constant. It is
For the purposes of indefinite integrals of this equation is correct, since the functions and are both antiderivatives of the function. Would you consider this expression as a definite integral with limits, so would the average ( the integral -free ) term omitted, because it is.
Logarithm
If only one term in the integrand, whose primitive function without table value is not close easily, you can occasionally integrate by parts by inserting the factor. This works for example with the logarithm. To determine the antiderivative of, is differentiated in the partial integration of the logarithm and is formed by the one- function primitive function. It is therefore
Product of sine and cosine
Sometimes you can make use of the fact that the original integral recurs after several steps of integration by parts on the right side of the equal sign, which then you can summarize by equivalence transformation with the original integral on the left side.
As an example, the indefinite integral
Calculated. Effected by exposing and, we obtain
And one obtains
Adding up on both sides of the equation, the output integral follows
If now divided on both sides by 2, we obtain
One exchanges in the partial integration, the roles of and, we obtain analogously
Because you can see that there are no excellent antiderivative.
Product of polynomial and exponential
Some indefinite integrals, it is advisable to choose a term that is not or only marginally changed in the integration, such as the natural exponential and trigonometric functions.
As an example, the indefinite integral
Considered. If, in each partial integration step and for the remaining term under the integral, we obtain
Cracked rational function
To determine the antiderivative of the integral representation of the arctangent is considered and this partial integration. It is then
For the antiderivative of then follows by term transformation
Derivation
The product rule of differential calculus states that for two continuous differentiable functions and the equality
Applies and follows by term transformation
Using the fundamental theorem of integral and differential calculus follows
Hence the rule
Results for partial integration.
Multidimensional integration by parts
The partial integration in multiple dimensions is a special case of Green's theorem: Let compact with smooth boundary sections. The edge is oriented by an outward normal unit field. Further, let a continuous differentiable vector field on an open area of and a steadily differentiable scalar field on. Then we have
With the abbreviation. Then follows the generalization of the partial integration in multiple dimensions
Rule of integration by parts for Stieltjesintegrale
There are two functions and of finite variation, then applies
Or written differently
Weak dissipation
In theory, the partial differential equations is a generalization of the derivative of a differentiable function was found by the method of partial integration.
Looking at one on an open interval ( classically ) differentiable function and an infinitely differentiable function with compact support in, then applies
Here, the integration by parts was used. Of boundary term, that is the integral term, without missing, just as the function, and therefore has a compact support and apply.
If the function is now chosen as a function, then, even if not differentiable (more precisely, no differentiable representative in the equivalence class has ), there exist a function, the equation
Met for each function. Such a function is called a weak derivative of. The resulting amount of weakly differentiable functions is a vector space and it belongs to the class of Sobolev spaces. The smooth functions with compact support whose vector space is designated, called test functions.
However, there is no function with the required condition, it can always be found a distribution so that the above condition is satisfied in the distribution sense. Then is called the distribution of derivative.