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ILATE Full Form, What Is ILATE Rule, Derivation of ILATE Rule, Important Points

ILATE Full Form : The ILATE full form is Inverse, Logarithmic, Algebraic, Trigonometric, and Exponential. It is used while performing integration by parts. ILATE is a rule used in integration when solving integration by product.
authorImageAnkit 3 Nov, 2023
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ILATE Full Form

ILATE Full Form : The ILATE full form is Inverse, Logarithmic, Algebraic, Trigonometric, and Exponential. It is used when performing integration by parts. We take the help of the ILATE rule in math when deciding which parts to integrate first when integrating a product element. There are two parts of a function in the product: u and dv. Now, we need to find its integration.

Check : JEE Main 2024 Exam

Now, we decide which one of u and dv will integrate first based on the ILATE rule. Read the complete article to learn more about the ILATE Full Form .

What is ILATE Full Form in Mathematics?

What is ILATE Full Form in Mathematics? : ILATE stands for Inverse, Logarithmic, Algebraic, Trigonometric, and Exponential. It is a rule used in the integration process when finding the integration for two elements in a product. The preference order, according to the rule of ILATE, is

ILATE In Maths Preference Order

There are five major elements which decide the order of integration when performing integration by parts. Check for more details about all these elements according to their preference.
  1. Inverse trigonometric functions: Any inverse function of trigonometry is given the first preference when performing data integration by parts. The inverse functions are sin inverse, cos inverse, etc.
  2. Logarithmic Functions: At the second number, logarithmic functions are given preference.
  3. Algebraic Functions: At the third number, algebraic functions are given preference.
  4. Trigonometric functions: The trigonometric functions such as sinx, cosx, tanx, etc. At the fourth number, trigonometric functions are given preference.
  5. Exponential Functions: At the third number, algebraic functions are given preference. At the fifth number, exponential functions are given preference.
Let us consider an example. We need to find the integration of the xsinx. We will use integration by parts. ∫ x sin x dx Now, here, we will use the ILATE rule to decide which part will be executed first. ∫ x sin x dx I     II Now, according to the ILATE rule, the first elements, u = x and dv = sinx are chosen. According to the format, algebraic functions are more preferred than logarithms. ∫ x sin x dx  =  -xcosx -  ∫  (--cosx)dx = -xcosx  + ∫ cosxdx Now, our problem reduces to cosx, which is easy to solve.

Examples Of ILATE Elements

In the table given below are some examples of the ILATE elements.
ILATE Rule Elements
ILATE Rule Elements Full Name Examples
I Inverse trigonometric functions \sin ^{-1} x, \cos ^{-1} x , etc.
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

How to Apply ILATE Rule

We apply the ILATE rule when we are performing integration by parts. Applying the ILATE rule is an easy task. However, one must remember the preference order of ILATE.
  • First, check the two functions f(x) and g(x) and mark them based on their type. Mark t (trigonometric), L (logarithmic), A (Algeabric), E (exponential) , I (inverse trigonometric).
  • Now, mark them first or second based on the ILATE rule. The function which appears first in the ILATE is given more preference and kept as the first function f(x).
  • After deciding the f(x) and g(x), apply the given formula to find the integration of the function.
∫f(x) g(x) dx = f(x)∫g(x)dx – ∫[f'(x)∫g(x)dx]dx

ILATE Full Form

ILATE Full Form : The full form of ILATE is Inverse, Logarithmic, algebraic, Trigonometric, and Exponential. Let us now look at some of the significant and essential points for ILATE.
  • ILATE is a rule used in integration when solving integration by product.
  • The ILATE rule is used to pick an appropriate element from the two available elements in the product.
  • If there are inverse functions such as sin inverse x, cos inverse x, etc., they are found here.

Check : NTA Updated JEE Main Syllabus Reduced 2024

Derivation of ILATE Rule

Let us now check the important formula for integration. Let us assume two elements, u and v, are differentiable functions of a single variable x. Then
Derivation of ILATE Rule
d/dx(uv) = u(dv/dx) + v(du/dx) By integrating both sides, we get; uv = ∫u(dv/dx)dx + ∫v(du/dx)dx or ∫u(dv/dx)dx = uv-∫v(du/dx)dx  ………….(1) Now let us consider, u=f(x) and  dv/dx = g(x) Thus, we can write now; du/dx = f'(x) and v = ∫g(x) dx Therefore, now equation 1 becomes; ∫f(x) g(x) dx = f(x)∫g(x) dx – ∫[∫g(x) dx] f'(x) dx or ∫f(x) g(x) dx = f(x) ∫g(x)dx – ∫[f'(x)∫g(x)dx]dx
∫f(x) g(x) dx = f(x)∫g(x)dx – ∫[f'(x)∫g(x)dx]dx The formula given above is used to solve integration by parts. We decide f(x) and g(x) based on the ILATE rule.

Check : JEE Main Maths Notes 2024

ILATE Rule Important Points

Candidates must keep certain things in mind while using the ILATE rule.
  • We decide on the first function for integration by parts using the ILATE Rule.
  • Select the first function according to the ILATE rule.
  • If there is only a single function, assume the second function as 1 in the case of logarithmic or inverse trigonometric functions.
  • We can use the ILATE rule multiple times while performing integration.

Also Check : JEE Main Marks vs Percentile

Full Form of ILATE FAQs

Q1. What is the rule of integration for product integration?

Ans: The product integration generally follows the following rule ∫ f(x) g(x) dx = f(x) ∫ g(x)dx – ∫[f'(x)∫g(x)dx] dx.

Q2. When do we need to use the ILATE Rule in math?

Ans: ILATE Forumla is used when calculating any product values present in product form. It is used to decide which one to get first.

Q3. What is the full form of ILATE?

Ans: The full form of ILATE is Inverse, Logarithmic, Algebraic, Trigonometric, and Exponential.

Q4. How do I use the ILATE rule?

Ans: ILATE is used when we need to perform integration by parts. We select one of the parts as the first preference and the other as the second. Check out the article to learn more about the ILATE rule.
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