Integrals are used in mathematics to find many useful quantities such as areas, volumes, displacement, and so on. The calculation of an integral is called integration which is represented by “∫”. When we generally talk about integrals, we usually mean definite integrals. Antiderivatives are represented by indefinite integrals. Apart from differentiation, integration is one of the two major calculus topics in mathematics (which measure the rate of change of any function with respect to its variables). It is a broad topic that is covered in higher-level classes such as Class 11 and 12. The integration by parts and by substitution is explained in detail.

**Definition of Integration**

The term integration refers to the summation of discrete data. The integral is computed to find the functions that will describe the area, displacement, and volume that occur as a result of a collection of small data that cannot be measured singularly. In a broad sense, the concept of limit is used in calculus where algebra and geometry are implemented. Limits aid in the investigation of the outcomes of points on a graph, such as how they get closer to each other until their distance is nearly zero. We are aware that there are two types of calculus –

- Differential Calculus
- Integral Calculus

**The concept of integration evolved to address the following types of issues:**

1. When the derivatives of the problem function are given, the problem function must be found.

2. To find the area bounded by a function’s graph under certain constraints.

These two problems resulted in the development of the concept known as “Integral Calculus,” which is made up of definite and indefinite integrals. The Fundamental Theorem of Calculus connects the concepts of differentiating a function and integrating a function in calculus.

**History of Integration**

Brook Taylor, who also proposed the famous Taylor’s Theorem, proposed the idea of integration by parts in 1715. In general, integrals are computed for functions that have differentiation formulas. Here Integration by parts is a technique for determining the integration of the product of functions that is also known as partial integration. It converts the integration of a function’s product into integrals for which a solution is easily computed.

**What is Integration By Parts in Integration?**

The formula is divided into two parts in the integration by parts method, and we can see the derivative of the first function named f(x) in the second part and the integral of the second function named g(x) in both parts. For clarity, these functions are frequently denoted by the letters ‘u’ and ‘v’. To know more about integration and to know about the other methods used in integration you can visit Cuemath.

**Integration By Parts Method**

- If the integrand function can be represented as a multiple of two or more functions, the Integration by Parts method can be used to integrate any given function.
- Consider an integrand function equal to f(x)g (x).f(x) g(x).
- In mathematics, the ILATE rule is used to select the first and second functions in this method.
- In mathematics, here’s how integration by parts is represented, see the format given below.

∫f(x) g(x).dx = f(x).∫g(x).dx–∫(f′(x).∫g(x).dx).dx

Which can be expressed as the integral of the product of any two functions = (First function × Integral of the second function) – Integral of [(differentiation of the first function) ×Integral of the second function]

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### What is the LIATE rule used in the Integration by Parts Method?

LIATE is known to be a rule which helps to decide which term should you differentiate first and which term should you integrate first.

- L stands for Logarithm
- I stands for Inverse
- Stands for Algebraic
- T stands for Trigonometric
- Estands for Exponential

The term which is closer to the letter L is differentiated first and the term which is closer to E is integrated first.