Which of the following does NOT belong in the category of integrals evaluated using the ILATE rule?

The ILATE rule is a mnemonic device used to help remember the order of functions to consider when choosing a substitution for integration by parts. The acronym stands for:

• I: Inverse trigonometric functions

• L: Logarithmic functions

• A: Algebraic functions

• T: Trigonometric functions

• E: Exponential functions

According to this rule, when faced with an integral that can be solved using integration by parts, one should prioritize functions in the order of inverses, logarithmic, algebraic, trigonometric, and exponential.

In this context, square root functions typically fall under the category of algebraic functions. They are not classified separately because they represent a specific case of algebraic functions (i.e., ( x^{1/2} )). Therefore, they do not belong to the list represented by the ILATE rule as they do not directly qualify as logarithmic, trigonometric, or exponential.

Logarithmic functions, exponential functions, and trigonometric functions all fit within the categories specified by the ILATE rule. Therefore, the correct response identifies square root functions as the one that does not fit into the integrals commonly evaluated using the ILATE rule.