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.e^x dx.

Solution:

Let’s choose x as the I-function and e^x as the II-function.

Apply the ILATE rule formula:

∫I × II dx = I ∫II dx – ∫I’ (∫II dx) dx

Therefore,

∫x.e^x dx

= x ∫e^x dx – ∫(1 ∫e^x dx) dx.

Now, we integrate ∫e^x dx to get e^x

∫xe^x dx = xe^x – ∫(1 ∫e^x dx) dx.

Integrate ∫(1 ∫ex dx) dx to get e^x

So,

∫x.e^x dx = x.(e^x – 1) + c

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.