This article will delve into the **LIATE** rule, offering a comprehensive understanding of its significance, application, and how it contributes to easing the **complex integration** process.

## Defining LIATE

**LIATE** is an acronym that stands for **Logarithmic**, **Inverse trigonometric**, **Algebraic**, **Trigonometric**, and **Exponential** functions. It is a rule of thumb to help you choose the “u” function when using the **integration by parts** formula.

The **integration by parts** formula is an integral analog to the product rule for differentiation. It is given as follows:

∫u dv = u v – ∫v du

When applying the **integration by parts** formula, we typically have to choose the functions ‘u’ and ‘dv’ from a given integrand that is a product of two functions. The choice of ‘u’ can greatly simplify the integration process. That’s where the **LIATE** rule comes in.

According to the **LIATE** rule, one should choose ‘u’ from the given integrand in the following** priority order**:

**L:**Logarithmic functions, such as**ln(x)****I:**Inverse trigonometric functions, such as**arcsin(x)**,**arccos(x)**,**arctan(x)****A:**Algebraic functions, like**polynomials**or**root functions****T:**Trigonometric functions, such as**sin(x)**,**cos(x)**,**tan(x)**,**cot(x)****E:**Exponential functions, such as**eˣ**,**aˣ**

Like all rules of thumb, **LIATE** is not a hard-and-fast rule. There are exceptions where the choice of **‘u’** may deviate from this order to simplify the **integral**. But in most cases, it proves to be a helpful guide.

Figure-1.

**Properties of LIATE Rule**

The **LIATE** rule, as a mnemonic device, helps mathematicians remember the priority order for selecting the ‘**u**‘ function when using the method of **integration by parts**. **LIATE** stands for **Logarithmic**, **Inverse trigonometric**, **Algebraic**, **Trigonometric**, and **Exponential** functions. The properties or aspects related to each part of **LIATE** are as follows:

**L (Logarithmic Functions)**

These functions are based on **logarithms**. The **natural logarithm** function ln(x) is the most commonly used, but others, such as **log base 10** or any other base, can also be involved. In the context of the **LIATE** rule, if a **logarithmic function** is a factor of the **integrand** in an integration problem, it’s typically a good candidate to select as ‘**u**‘ in the integration by parts formula.

**I (Inverse Trigonometric Functions)**

Inverse trigonometric functions, such as **arcsin(x)**, **arccos(x)**, **arctan(x)**, **arccsc(x)**, **arccot(x)**, and **arcsec(x)**, return the angle for a given ratio in a right triangle. In the **LIATE** rule, if there is an **inverse trigonometric** function in the integrand, it takes the second priority after **logarithmic functions** to be chosen as ‘**u**‘ in the **integration by parts** formula.

**A (Algebraic Functions)**

**Algebraic functions** are any function that polynomial expressions can define. They include **polynomials**, **root functions**, and **rational functions**. In the context of the **LIATE** rule, an algebraic function should be chosen as ‘**u**‘ only if there are no logarithmic or inverse trigonometric functions in the **integrand**.

**T (Trigonometric Functions)**

**Trigonometric functions** are based on the ratios of the sides of a right triangle to its angles. They include **sin(x)**, **cos(x)**, **tan(x)**, **cot(x)**, **sec(x)**, and **csc(x)**. In the **LIATE** rule, a trigonometric function should be chosen as ‘**u**‘ only if there are no **logarithmic**, **inverse trigonometric**, or **algebraic functions** in the **integrand**.

**E (Exponential Functions)**

**Exponential functions** have a variable in the exponent, like **eˣ** or **aˣ**, where ‘a’ is a constant. According to the **LIATE** rule, these functions are usually the last to be selected as ‘**u**‘ in the integration by parts formula.

**Ralevent Formulas **

The** LIATE** rule is a heuristic used to determine the order of functions in the **integration by parts** formula. Here’s a reminder of the **LIATE** rule:

**L**:**Logarithmic**functions, such as**ln(x)****I**: Inverse**trigonometric**functions, such as**arctan(x)**or**arcsin(x)****A**:**Algebraic**functions, like polynomials**x²**or radicals**√x****T**:**Trigonometric**functions, like**sin(x)**,**cos(x)**, etc.**E**:**Exponential**functions, such as**eˣ**,**aˣ**

The basic formula for integration by parts is:

∫u dv = u v – ∫v du

When we have a **product of functions** to integrate, we choose which function to be **“u”** and which to be **“dv”** based on the **LIATE** rule. The function that comes first in the **LIATE** rule is chosen as **“u”** while the remaining function is **“dv.”** Then we find **“du”** by **differentiating “u”** and** “v”** by integrating** “dv.”**

In the context of the **LIATE** rule, there isn’t a specific formula. However, it’s a method to apply the formula for integration by parts. The **LIATE** rule helps determine the choice of **“u”** and **“dv”** when applying **integration by parts**.

For example, if we want to integrate **x * eˣ dx**, following the LIATE rule, we’d choose **u = x** (Algebraic) and **dv = eˣ dx** (Exponential). Then **du = dx**, **v = eˣ**, and the integration by parts formula give:

∫x eˣ dx = x eˣ – ∫eˣ dx

∫x eˣ dx = x eˣ – eˣ + C

The rule doesn’t always work perfectly, and sometimes you might need to apply **integration by parts** more than once or use a different method. But it’s a helpful guideline when first setting up the **integral**.

**Exercise **

**Example 1**

Problem: ∫x ln(x) dx

Figure-2.

### Solution

Here, let **u = ln(x)** (L in LIATE) and **dv = x dx** (A in LIATE). Then, **du = dx/x** and **v = 0.5 * x²**. Using the formula for integration by parts, ∫u dv = u v – ∫v du, we get:

∫x ln(x) dx = 0.5x² ln(x) – ∫0.5x dx

∫x ln(x) dx = 0.5x² ln(x) – 0.25x² + C

**Example 2**

Problem: ∫x eˣ dx

### Solution

Here, let **u = x** (A in LIATE) and **dv = eˣ dx** (E in LIATE). Then, **du = dx** and **v = eˣ**. Applying the integration by parts formula, we get:

∫x eˣ dx = x eˣ – ∫eˣ dx

∫x eˣ dx = x eˣ – eˣ + C

**Example 3**

Problem: ∫x sin(x) dx

Figure-3.

### Solution

Here, let **u = x** (A in LIATE) and **dv = sin(x) dx** (T in LIATE). Then, **du = dx** and **v = -cos(x)**. Applying the integration by parts formula, we get:

∫x sin(x) dx = -x cos(x) – ∫-cos(x) dx

∫x sin(x) dx = -x cos(x) + sin(x) + C

**Example 4**

Problem: ∫x² ln(x) dx

### Solution

Here, we will use integration by parts twice. First, let **u = ln(x)** (L in LIATE) and **dv = x² dx** (A in LIATE). Then, **du = dx/x** and** v = (1/3) * x³**. Applying the integration by parts formula, we get:

∫x² ln(x) dx = (1/3) * x³ ln(x) – ∫(1/3) * x² dx

∫x² ln(x) dx= (1/3) * x³ ln(x) – (1/9) * x³ + C

**Example 5**

Problem: ∫x² eˣ dx

Figure-4.

### Solution

Here, we will also use integration by parts twice. First, let **u = x²** (A in LIATE) and **dv = eˣ dx** (E in LIATE). Then, **du = 2 * x dx** and** v = eˣ**. Applying the integration by parts formula, we get:

∫x² * eˣ dx = x² eˣ – ∫2 * x eˣ dx

Now, we will use integration by parts for the second integral: Let u = 2x (A in LIATE) and dv = eˣ dx (E in LIATE). Then, du = 2 dx and v = eˣ. Again applying the formula, we get:

∫2 * x eˣ dx = 2x eˣ – ∫2 e^x dx

∫2 * x eˣ dx = 2x eˣ – 2 eˣ + C

Substituting this back into the first integral, we get:

∫x² eˣ dx = x² eˣ – (2x eˣ – 2 eˣ) + C

∫x² eˣ dx = (x² – 2x + 2) * eˣ + C

**Example 6**

Problem: ∫ln(x)/x dx

### Solution

Here, let **u = ln(x)** (L in LIATE) and **dv = dx/x** (A in LIATE). Then, **du = dx/x** and **v = 1**. Applying the integration by parts formula, we get:

∫ln(x)/x dx = ln(x) – ∫dx

∫ln(x)/x dx = ln(x) – x + C

**Example 7**

Problem: ∫x arctan(x) dx

### Solution

Here, let** u = arctan(x)** (I in LIATE) and **dv = x dx** (A in LIATE). Then, **du = dx/(1 + x²)** and **v = 0.5 * x²**. Applying the integration by parts formula, we get:

∫x arctan(x) dx = 0.5 * x² * arctan(x) – ∫0.5 * x² dx/(1 + x²)

This integral would further require partial fraction decomposition to solve completely.

**Example 8**

Problem: ∫x ln(x) eˣ dx

### Solution

This is a c**omplex problem** that would require using integration by parts multiple times and applying the **LIATE** rule appropriately. Each integration by parts would reduce the degree of the algebraic part until it becomes a simple function. It’s beyond the scope of this brief explanation, but it’s a good exercise to work through!

**Applications **

**Engineering**

Many **engineering problems** involve the evaluation of integrals that arise in the context of physical phenomena. The **LIATE** rule can be particularly useful for performing these integrations. For instance, in **Electrical Engineering**, the circuits’ voltage and current calculations often require the integration of functions that can be solved using the **LIATE** rule.

**Mathematics**

By following the **LIATE** rule, mathematicians can often select the most appropriate functions for **differentiation** and **integration**, leading to more efficient and successful integration by parts computations.

However, it’s important to note that the **LIATE** rule is a guideline, not a strict one. There may be cases where a different order of functions works better depending on the problem.

**Physics**

The **LIATE** rule is often used in **Physics** to solve various problems, especially in **Quantum Mechanics**, where wave functions often need to be integrated to find probabilities. It is also used in **Classical Mechanics** for solving problems related to **motion**, **energy**, and** momentum**.

**Economics and Finance**

In these fields, the **LIATE** rule can solve integrals in models involving continuous time, such as certain **economic growth models** or **option pricing models** in finance.

**Biology and Medicine**

The **LIATE** rule can help integrate functions that model **biological systems** or processes, such as **drug concentration** in the bloodstream over time or the **population dynamics** of species.

**Computer Science**

In **algorithm analysis**, integrals often appear when dealing with continuous approximations of discrete problems. The **LIATE** rule can be used to solve such integrals, which might involve, for example, the **running time of an algorithm** as a function of the input size.

By applying the **LIATE** rule, researchers can select appropriate functions for** integration** and **analyze** the efficiency and behavior of algorithms in a continuous context. This aids in understanding the scalability and performance of algorithms as the input size increases.

**Statistics and Data Science**

In **statistics**, many problems involve the computation of **expectations**, **variances**, and other **moments**, which require integration. The **LIATE** rule is often used to solve these kinds of problems.

*All images were created with GeoGebra.*