ILATE rule stands for Inverse Trigonometric Function, Logarithmic Function, Algebraic Function, Trigonometric Function, and Exponential Function. It tells about the priority order in which functions are selected for their integration. It is an important concept in solving integration problems.
This formula is also called the 'uv integration formula'. If we have to find the integration of a function that is a product of two functions then we use the ILATE rule of integration. In this article, we will learn about, What the is ILATE Rule, How to Apply the ILATE Rule, ILATE Rule Examples, and others in detail.
What is ILATE Rule?
The ILATE Rule, the acronym for Inverse, Logarithmic, Algebraic, Trigonometric, and Exponential functions, guides the order in which different functions are prioritized during integration or differentiation. The ILATE rule is an acronym commonly used in integral calculus to determine which function should be selected as u and which as dv when employing the integration by parts method.
The ILATE rule is commonly employed when integrating products of two functions where standard integration techniques like substitution or simpler rules do not apply. It serves as a useful strategy for choosing u and dv to simplify the integral in cases involving products of functions.
ILATE stands for
- Inverse functions: Such as logarithmic or exponential functions
- Linear functions: Including polynomials or functions involving algebraic operations
- Algebraic functions: Another reference to algebraic functions
- Transcendental functions: Such as sine, cosine, tangent, etc.
- Exponential functions: Like ex
Using ILATE rule we first find I and II function then we use the formula i.e.
= ∫ (First Function).(Second Function).dx
= First Function ∫ (Second Function) dx - ∫ [ d/dx (First Function) ∫ (Second Function dx) ] dx
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How to Apply ILATE Rule?
To apply the ILATE Rule, identify the functions in your expression and follow the sequence: Inverse, Logarithmic, Algebraic, Trigonometric, and Exponential. Choose the function with the highest priority as the first one to differentiate or integrate.
Why is ILATE Rule Used?
ILATE Rule is used to streamline the process of integration and differentiation by providing a systematic approach. It ensures efficient handling of different functions, minimizing errors and simplifying complex calculations. The ILATE rule is significant in integration by parts as it offers a systematic approach to selecting which function to assign as u and which to assign as dv. Its importance lies in simplifying the integration process by providing a guideline for prioritizing functions based on their rate of growth or simplification upon differentiation or integration.
By following the ILATE rule, one can make more informed decisions about the assignment of u and dv, aiming to simplify the integral by strategically choosing the function for differentiation that reduces faster or more effectively through repeated differentiation compared to the other function. This strategy often results in integrals that are easier to solve or require fewer iterations, streamlining the overall integration process.
Applying ILATE Rule Using I and II Functions
The ILATE (Inverse, Logarithmic, Algebraic, Trigonometric, Exponential) rule is a method used for integrating products of functions. To apply this rule, identify the functions in your integral as I and II.
- I-Function: This is the part of the integrand that you choose to differentiate.
- II-Function: This is the remaining part that you choose to integrate.
This can be easily understood by the example added below,
Example: Solve ∫x.ex dx.
Solution:
Let's choose x as the I-function and ex as the II-function.
Apply the ILATE rule formula:
∫I × II dx = I ∫II dx - ∫I' (∫II dx) dx
Therefore,
∫x.ex dx
= x ∫ex dx - ∫(1 ∫ex dx) dx.
Now, we integrate ∫ex dx to get ex
∫xex dx = xex - ∫(1 ∫ex dx) dx.
Integrate ∫(1 ∫ex dx) dx to get ex
So,
∫x.ex dx = x.(ex - 1) + c
ILATE Rule for Single Function
For a single function, ILATE Rule is still applicable. Identify the nature of the function and follow the sequence: Inverse, Logarithmic, Algebraic, Trigonometric, and Exponential.
When dealing with a single function, apply the ILATE rule by prioritizing functions in the order of Inverse, Logarithmic, Algebraic, Trigonometric, and Exponential.
Example: ∫ln(x) dx.
Solution:
∫1.ln(x) dx
In this case, ln(x) is a logarithmic function and 1 is algebraic function
According to ILATE, logarithmic functions take precedence over algebraic functions. So,
- First Function: ln(x)
- Second Function: 1
∫ 1.ln x 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
= x(ln x - 1) + C
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Solved Examples on ILATE Rule
Example 1: Solve the integral: ∫xcos(x)dx
Solution:
∫x.cos(x).dx
Let,
- u(I function) = x (Algebraic Function)
- v(II function) = sin x (Trigonometric Function)
= ∫x.cos(x).dx
= x{∫(cos x)dx} - ∫sin x (dx/dx).dx
= xsin(x) − ∫sin(x).dx
= xsin(x) + cos(x) + C
Example 2: Solve the integral: ∫ex ln(x)dx
Solution:
∫ex ln(x)dx
Let,
- u(I function) = ln(x) (Logarithmic Function)
- v(II function) = ex (Exponential Function)
Apply Integration by Parts
du = 1/x dx, v = ex
Now,
∫exln(x) dx = exln(x) - ∫ 1/xex dx
= exln(x)dx - ∫ 1/x d(ex)
= exln(x)dx - ex/x + C
Example 3: Solve the integral: ∫x ln(x)dx
Solution:
∫x ln(x)dx
Let,
- u(I function) = x (Algebraic Function)
- v(II function) = ln(x) (Logarithmic Function)
Apply Integration by Parts
= ∫x ln(x) dx
= (1/2) × x2 × ln(x) - ∫(1/2) × x dx
= (1/2)x2ln(x) - (1/4)x2 + C
Example 4: Solve the integral: ∫16xcos(x)dx
Solution:
∫16xcos(x)dx
Let,
- u(I function) = x (Algebraic Function)
- v(II function) = cos(x) (Trigonometric Function)
Apply Integration by Parts
u = x, dv=cos(x)dx
du = dx, v=sin(x)
Now,
= 16∫ x cos(x) dx
= 16 x sin(x) - ∫ 16sin(x) dx
= 16x sin(x) + 16cos(x) + C
Example 5: Solve the integral: ∫xex dx
Solution:
∫xex dx
Let,
- u(I function) = x (Algebraic Function)
- v(II function) = ex (Exponential Function)
Apply Integration by Parts
= ∫x.ex dx
= x.ex - ∫1.ex dx
= xex - ex + C
Practice Questions in ILATE Rule
Q1: Solve the integral: ∫ x sin(x)dx.
Q2: Solve the integral: ∫x3 ln(x)dx.
Q3: Solve the integral: ∫e2x ln(x)dx.
Q4: Solve the integral: ∫4exln(x)dx
Q5: Solve the integral: ∫ x2 cos(x)dx.
Summary
The ILATE rule is a mnemonic device used to choose the order of functions for integration by parts, which is a technique to integrate products of functions.
When applying integration by parts, you typically choose uuu (the function to differentiate) from the higher priority functions in the ILATE rule and dv (the function to integrate) from the lower priority functions. This choice generally simplifies the integration process by ensuring that the derivative of uuu and the integral of dvdvdv are easier to handle. For instance, in the integral \int x e^x dx , choosing u=x (algebraic) and dv=exdx (exponential) simplifies the integration process.