ILATE Full Form
What is the full form of ILATE?
In the ILATE rule, each letter stands for a particular kind of function: I for inverse trigonometric, L for logarithmic, A for algebraic, T for trigonometric, and E for exponential. The most beneficial rule for part-by-part integration is the ILATE rule. This rule determines which function should be used as the first function when the parts do the integration. This rule can also be substituted with the LIATE rule.
Let's explore the ILATE rule and its applications in greater detail, with more examples and solutions.
1. What is the ILATE (LIATE) rule?
The most popular rule in the process of integration by parts is called the ILATE rule, and it makes choosing the first function and the second function very simple. The formula for "integration by parts" can be written in one of two ways:
\int u \mathrm{dv}=u v-\int v \mathrm{~d} u
∫ (first function) (second function) dx = first function ∫ (second function) dx - ∫ [ d/dx (first function) ∫ (second function dx) ] dx
The words "first" and "second" are used in this formula. It implies that the order of the functions has some significance in the particular product of functions. The first function (u) is typically chosen in a way that makes it simple to calculate the integral of its derivative. We employ the ILATE rule to make the choice of the first function more straightforward. This rule is used to determine the priority of the first function. The first function should be chosen so that it appears as high as possible on the list above. Each letter in the ILATE rule represents the abbreviation of a particular kind of function described as follows:
ILATE Rule letters | Abbreviations | Examples |
I | Inverse trigonometric functions | \sin ^{-1} x, \cos ^{-1} x
|
L | Logarithmic functions | \log x, \ln x
|
A | Algebraic functions | x^2, \sqrt{x}
|
T | Trigonometric functions | \sin x, \cos x
|
E | Exponential functions | e^x, 2^x
|
, etc.
2. How to Implement the ILATE rule?
We are aware that in order to combine the results of two various types of functions, we use integration by parts. We must determine the first function before applying integration by parts, and we can do this by using the ILATE rule. Simply choose the function that appears first (from the top) in the above/below list to serve as the initial function.
The following provides a detailed explanation of the ILATE rule application steps:
Assign each function one of the following types: Inverse trigonometric (I), Logarithmic (L), Algebraic (I), Trigonometric (T), or Exponential (E).
Select the function that is listed first in the ILATE (or) LIATE order of the functions.
The second function should be the one that is left.
Following that, use the integration by parts formula.
3. What is the "Integration by Parts" formula?
The method of integration known as "integration by parts" is quite advantageous when two functions are multiplied together, but it also has other applications. Although there are many examples, it is crucial to first define the rule:
\int u v d x=u \int v d x-\int u^{\prime}\left(\int v d x\right) d x
Where
U = function u(x)
V = function v(x)
U’ = derivative of the function u(x)
This approach is very helpful. This method is employed to achieve the desired outcomes in several situations where students are required to integrate the product of two functions.
4. How do I apply the ILATE rule to a single function?
There is no direct integration rule that can be used to find the integrals of single functions like ln x,sin-1x, etc. when they need to be integrated. Even though we can find such rules, they are hard to recall. When this occurs, we add "times 1" after the given function so that the integrand has two functions and the LIATE rule can be applied.
5. Important Pointers for the ILATE Rule:
In an integration-by-parts process, the first function is determined using the ILATE rule.
Select the first function from the list provided by the ILATE rule, starting at the top.
In the case of logarithmic and inverse trigonometric functions, the ILATE rule can also be used to integrate a single function by writing the second function as 1.
The LIATE rule operates similarly to the ILATE rule.
The ILATE rule can be used multiple times during integration.
Sometimes the process is made simple by selecting integration strategies other than the ILATE rule. It is simpler to use the substitution method than the integration by parts, for instance, to find (ln x)/x dx.
Frequently Asked Questions (FAQs)
ILATE and LIATE are also equivalently correct.
The ILATE rule merely specifies the order in which preference to the choice of functions should be given, so if it is not followed, integration by parts will still produce the correct result.
No, using the integration by parts is actually not necessary.
The main applications of integration are computing the volumes of three-dimensional objects and determining the areas of two-dimensional regions. Therefore, determining the area away from the X-axis on the curve is equivalent to determining the integral of a function with respect to x.
Mathematician Brook Taylor first proposed the concept of integration by parts in 1715. For the Riemann-Stieltjes and Lebesgue-Stieltjes integrals, more general formulations of integration by parts are available. Summation by parts is used to represent sequences in discrete mathematics.