ILATE Full Form, ILATE Rule in Integration, Preference Order, Formula and Examples

ILATE Full Form: In the study of calculus, students generally get confused while doing integration by parts. Most of the time, two functions get multiplied and students get stuck in choosing which one to differentiate and which one to integrate. To end this confusion, mathematicians introduced a rule called the ILATE rule. The ILATE Full Form in Mathematics is Inverse, Logarithmic, Algebraic, Trigonometric and Exponential.

This rule can help you in selecting which function to choose as u and which to choose as dv while using the integration by parts method. It has a fixed preference order of I > L > A > T > E, which is written as Inverse > Logarithmic > Algebraic > Trigonometric > Exponential. It means that Inverse functions have the highest preference and Exponential functions have the least preference. 

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What is ILATE Full Form in Mathematics?

The ILATE Full Form in Mathematics stands for:

I – Inverse Functions
L – Logarithmic Functions
A – Algebraic Functions
T – Trigonometric Functions
E – Exponential Functions

This rule is followed when you are solving an integration by parts. Integration by parts is used when the question has two different types of functions multiplied together. The formula for integration by parts is:

∫u dv = u*v − ∫v du

For this formula, we need to choose u and dv smartly. Choosing the wrong option makes the question tougher. Use ILATE Full Form to make this choice effortlessly.

Example: For ∫x sin x dx, we have two categories of functions. x is algebraic and sin x is trigonometric. As per ILATE, Algebraic comes before Trigonometric (A > T), so u = x and dv = sin x dx.

When you use this logic in every integration, it saves time and makes your solution simpler.

Also Check: JEE Full Form

ILATE in Maths Preference Order

The ILATE in Maths Preference Order gives a sequence of priority to choose the function u while solving integration by parts. The order is:

If two functions are multiplied, the one that comes first in this order becomes u, and the other becomes dv.

For example, in ∫x e^x dx, we have Algebraic (x) and Exponential (e^x). According to ILATE, Algebraic has higher priority than Exponential, so u = x and dv = e^x dx.

Then, using the formula ∫u dv = u*v − ∫v du, we can easily solve it.

This simple order I > L > A > T > E helps students decide quickly without confusion.

Examples of ILATE Elements

Let us understand examples of each type of function covered in the ILATE Full Form in Mathematics.

Example 1

Find ∫x log x dx
Here, we have Algebraic (x) and Logarithmic (log x). According to ILATE, Logarithmic comes before Algebraic. So u = log x and dv = x dx.

Now du = (1/x) dx and v = x² / 2.
By formula: ∫u dv = u*v − ∫v du
= (x²/2)log x − ∫(x²/2)(1/x) dx
= (x²/2)*log x − ∫(x/2) dx
= (x²/2)*log x − (x²/4) + C

Example 2

Find ∫x e^x dx
Here, Algebraic (x) and Exponential (e^x) are given. According to ILATE, Algebraic comes before Exponential. So u = x and dv = e^x dx.

du = dx and v = e^x
∫u dv = uv − ∫v du
= xe^x − ∫e^x dx
= e^x(x − 1) + C

These examples show that following ILATE simplifies integration step by step.

How to Apply ILATE Rule

The steps involved in using the ILATE Rule are as follows: 

Step 1: In the given integral, identify both the functions. 

 Step 2: Locate their category from ILATE Full Form. 

 Step 3: Select the function that is earlier in the ILATE order as u. 

 Step 4: Select the other function as dv. 

 Step 5: Apply the formula ∫u dv = u*v − ∫v du.

Let’s take another example.

Find ∫x sin x dx
We have Algebraic (x) and Trigonometric (sin x).
According to ILATE, Algebraic comes before Trigonometric.
So u = x and dv = sin x dx.

Then du = dx and v = −cos x.
By formula: ∫u dv = u*v − ∫v du
= −x cos x + ∫cos x dx
= −x cos x + sin x + C

Hence, the final answer is −x cos x + sin x + C.

By following ILATE, choosing the correct u and dv becomes automatic.

ILATE Full Form

ILATE Full Form is Inverse, Logarithmic, Algebraic, Trigonometric and Exponential. This acronym works like a memory pal or a trick for students to easily memorize the order.

ILATE itself is not a theorem. It is a principle that is followed based on the nature of the functions after we differentiate or integrate them. Inverse and logarithmic are difficult to integrate, hence they are at the first position. Exponentials is easy to integrate, hence they are at the last position.

The integration questions which are solved by the ILATE Rule are the ones involving the product of two functions. It is always important to remember ILATE in such questions. ILATE works like a rule book for us to decide the right order.

For example, in ∫tan⁻¹x * e^x dx,
Inverse (I) vs Exponential (E) → Inverse comes first.
So u = tan⁻¹x and dv = e^x dx.

This approach works for all similar problems.

Derivation of ILATE Rule

The Derivation of ILATE Rule is based on the product rule of differentiation.

Start with the product rule:
d(uv)/dx = u*(dv/dx) + v*(du/dx)

Integrate both sides with respect to x:
∫ d(uv)/dx dx = ∫u*(dv/dx) dx + ∫v*(du/dx) dx

Simplify it:
uv = ∫u dv + ∫v du

Now rearrange:
∫u dv = u*v − ∫v du

This is the formula used in integration by parts.

But while solving, we must select which function will be u and which will be dv. If we choose u incorrectly, the process becomes complex. To avoid this problem, mathematicians created the ILATE order, which gives a logical way to decide.

Thus, the ILATE Rule is derived from the basic formula of integration by parts but gives a clear order to follow while selecting functions.

ILATE Rule Important Points

A Few Important Points regarding the ILATE Rule are listed below:

1. ILATE Rule helps you to decide u and dv while using the integration by parts formula.

2. Try to take u as per ILATE rule (I > L > A > T > E).

3. If only one function is there, then let u = that function and dv = 1 dx.

4. ILATE rule is more beneficial when a question has two different types of functions.

5. If both the functions belong to same category, then let u be the one which becomes simple when differentiated.

6. Inverse and logarithmic functions are always taken as u because their integration is not so easy.

7. ILATE Rule can be applied again and again if the integral is of a type that requires multiple steps of integration by parts.

Example of multiple use

Find ∫x² e^x dx
First, u = x², dv = e^x dx.
Then du = 2x dx and v = e^x.
Using formula: ∫u dv = u*v − ∫v du
= x² e^x − ∫2x e^x dx
Now, again apply ILATE for ∫x e^x dx.
Finally, you get e^x(x² − 2x + 2) + C.

This example shows how ILATE helps even in longer problems.