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ILATE Rule

ILATE rule is the most helpful rule used in integration by parts. This rule is used to decide which function is to be chosen as the first function when the integration is done by parts. Instead of this rule, LIATE rule can also be applied.

Let us learn more about ILATE rule and its applications in detail along with more solved examples.

1.	What is ILATE (LIATE) Rule?
2.	How to Apply ILATE Rule?
3.	How to Apply ILATE Rule for a Single Function?
4.	FAQs on ILATE Rule

What is ILATE (LIATE) Rule?

ILATE rule is a rule that is most commonly used in the process of integration by parts and it makes the process of selecting the first function and the second function very easy. The integration by parts formula can be written in two ways:

∫ u dv = uv - ∫ v du.
∫ (first function) (second function) dx = first function ∫ (second function) dx - ∫ [ d/dx (first function) ∫ (second function dx) ] dx

In this formula, we used the terms "first" and "second". It means there is definitely some importance for the order of the functions in the given product of functions. Usually, the first function (u) will be selected in such a manner that the process of finding the integral of its derivative must be easy. To simplify the selection of the first function, we use the ILATE rule. The priority of first function is done by using this rule. In ILATE rule, each letter stands for the abbrevation of a specific type of function described as follows:

Letters of ILATE Rule	Abbreviations	Examples
I	Inverse trigonometric functions	sin^-1x, cos^-1x, etc
L	Logarithmic functions	log x, ln x
A	Algebraic functions	x², √x
T	Trigonometric functions	sin x, cos x
E	Exponential functions	e^x, 2^x

The first function should be selected in such a way that it is as close to the top of the above list as possible.

How to Apply ILATE Rule?

We know that we use integration by parts to integrate the product of two different types of functions. To apply integration by parts, we have to decide what is the first function first and it can be done by using the ILATE rule. It is very simple, just select the function that comes at first (from the top) of the above/below list as the first function.

ILATE rule (or) LIATE rule

The steps of applying ILATE rule are explained in detail below:

Identify the type of each function as Inverse trigonometric (I), Logarithmic (L), Algebraic (I), Trigonometric (T), or Exponential (E).
See which of the given functions comes first in the order of ILATE (or) LIATE and choose it as the first function.
Select the remaining function as the second function.
Then apply the integration by parts formula.

For example, when we have to integrate ∫ x ln x dx, and if we apply the above steps:

x is the algebraic function; ln x is the logarithmic function.
Among Algebraic (A) and Logarithmic (L), we know that Logarithmic (L) comes first in ILATE rule. So the first function is ln x.
The remaining function, x, is the second function.
By applying the integration by parts formula:
∫ x ln x dx = (ln x) ∫ x dx - ∫ [d/dx (ln x) ∫ x dx] dx
= (ln x) (x²/2) - ∫ (1/x) (x²/2) dx
= (x² ln x)/2 - (1/2) ∫ x dx
= (x² ln x)/2 - (1/2) (x²/2) dx
= (x² ln x)/2 - (x²/4) + C

Here are a few more examples for the application of the first function using the ILATE rule.

∫ ln x sin x dx → ln x is the first function.
∫ x csc x dx → x is the first function.
∫ x sin^-1x → sin^-1x is the first function.
∫ x e^x dx → x is the first function.

How to Apply ILATE Rule for a Single Function?

Sometimes, we come across integrating single functions such as ln x, sin^-1x, etc but there is no direct integration rule available to find such integrals. Though we can find such rules, they are difficult to remember. In such cases, we write "times 1" after the given function, and then the integrand will have two functions, thereby the application of LIATE rule is possible. Consider the example below.

Example: Find the integral ∫ ln x dx.

Solution:

We write the given integral as, ∫ ln x × 1 dx.

By LIATE rule, it is clear that the first function is ln x and the second function is 1. Now, applying the integration by parts,

∫ ln x × 1 dx = (ln x) ∫ 1 dx - ∫ [d/dx (ln x) ∫ 1 dx] dx

= (ln x) x - ∫ (1/x) (x) dx

= x ln x - ∫ 1 dx

= x ln x - x + C

Important Notes in ILATE Rule:

ILATE rule is used to determine the first function in integration by parts.
Choose the first function as the function which comes in the list given by ILATE rule (from the top).
ILATE rule can be used to integrate a single function (in the case of logarithmic and inverse trigonometric functions) as well by writing the second function as 1.
LIATE rule works the same way as the ILATE rule.
We can apply the ILATE rule more than once in integration.
Sometimes, choosing other integration methods than the ILATE rule makes the process easy. For example to find ∫ (ln x) / x dx, using the substitution method is easier instead of using the integration by parts.

☛ Related Topics:

Examples on ILATE Rule

Example 1: Evaluate the integral ∫ (ln x) / x² dx.

Solution:

Here, ln x is the logarithmic function and 1/x² is algebraic. So by ILATE rule, first function = ln x and second function = 1/x² = x^-2.

Bu the integration by parts formula:

∫ (ln x) / x dx = (ln x) ∫ (x^-2) dx - ∫ [ d/dx (ln x) · ∫ x^-2 dx] dx
= (ln x) (-1/x) - ∫ (1/x) (-1/x) dx
= (- ln x) / x + ∫ x^-2 dx
= (- ln x) / x - 1/x + C

Answer: (- ln x) / x - 1/x + C.
Example 2: What is the value of the integral ∫ x sin x dx?

Solution:

Here, x is the algebraic function and sin x is the trigonometric function. So by using the LIATE rule, the first function = x and the second function = sin x. Applying integration by parts,

∫ x sin x dx = x ∫ sin x dx - ∫ [ d/dx (x) sin x ] dx
= x (- cos x) - ∫ 1 (- cos x ) dx
= - x cos x + ∫ cos x dx
= - x cos x + sin x + C

Answer: - x cos x + sin x + C
Example 3: Find the value of the integral ∫ e^x sin x dx.

Solution:

In the given integrand, e^x is the exponential function and sin x is the trigonometric function. By ILATE rule, sin x should be assumed as first and e^x should be assumed as the second function. Then by integration by parts,

∫ e^x sin x dx = sin x ∫ e^x dx - ∫ [ d/dx (sin x) ∫ e^x dx ] dx
= sin x (e^x) - ∫ e^xcos x dx

We have to apply the ILATE rule again for ∫ e^xcos x dx. Then we get:

∫ e^x sin x dx = e^x sin x - [ cos x ∫ e^x dx - ∫ [ d/dx (cos x) ∫ e^x dx ] dx ]
= e^x sin x - [ e^x cos x - ∫ (- sin x) e^x dx]
= e^x sin x - e^x cos x - ∫e^x sin x dx

If we assume ∫ e^x sin x dx = I, then we get

I = e^x sin x - e^x cos x - I

Adding I on both sides,

2I = e^x sin x - e^x cos x

Dividing both sides by 2,

I = (e^x sin x - e^x cos x) / 2

Thus, ∫ e^x sin x dx = (e^x sin x - e^x cos x) / 2 + C

Answer: (e^x sin x - e^x cos x) / 2 + C

Note: We have applied the ILATE rule twice. From this solution, we can derive the following two formulas:
- ∫ e^ax sin bx dx = e^ax/ (a² + b²) [a sin bx - b cos bx] + C
- ∫ e^ax cos bx dx = e^ax/ (a² + b²) [a cos bx + b sin bx] + C

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Practice Questions in LIATE Rule

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FAQs on ILATE Rule

What Does ILATE Rule Stand For?

Every letter in the ILATE rule stands for a specific type of function:

I : Inverse trigonometric function
L: Logarithmic function
A: Algebraic function
T: Trigonometric function
E: Exponential function

Is ILATE Rule Mandatory?

No, it is not mandatory to choose the first function in integration by parts. Generally, the first function is chosen in such a way that integrating its derivative is easy. But sometimes, using this logic to identify the first function is harder. In such cases, ILATE is useful, but again, it is NOT mandatory.

Which is Correct ILATE Rule or LIATE Rule?

Both ILATE and LIATE rules work the same way. These rules make the process of choosing the first function in integration by parts easier.

What is I in LIATE Rule Stand For?

Every single letter of LIATE rule stands for a type of function. 'I' stands for inverse trigonometric function.

How to Select the First Function Using ILATE Rule?

In the integration by parts formula, the first function "u" should be such that it comes first (when compared to the other function dv) in the list given by the ILATE rule from the top. For example, to integrate x² ln x, ln x is the first function as Logarithmic (L) comes first before the Algebraic (A) in the ILATE rule.

How to Use ILATE Rule?

ILATE rule stands for I (Inverse trigonometric), L (Logarithmic), A (Algebraic), T (Trigonometric), and E (Exponential). To apply this rule, just see which of the two given functions while applying the integration by parts comes first in the above-mentioned order.

Can we Apply LIATE Rule for a Single Function?

Yes, we can apply LIATE rule in case of integrating a single logarithmic/inverse trigonometric function. In this case, the second should be assumed as 1. For example, to integrate ∫ sin^-1x dx, write it as ∫ sin^-1x × 1 dx and then consider sin^-1x as the first function (u), 1 as the second function (dv), and then apply the integration by parts formula ∫ u dv = uv - ∫ v du.

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