Integration vs Partial Integration: Difference and Comparison

The solving of integral functions by using formula or partial methods is called Integration. Over and above, differentiation and integration are calculus’s two most fundamental, essential operations.

It acts as a tool to decipher problems in mathematics and physics; the area of a variable shape, the distance of a curve, and the volume of a solid.

Key Takeaways

1. Integration calculates the area under a curve or the antiderivative of a function, providing a way to find the accumulated value of a variable over a specific interval, whereas partial integration, also known as integration by parts, is a technique used to integrate products of two functions.
2. Integration is a fundamental concept in calculus, applicable to a wide range of problems in mathematics, physics, and engineering. In contrast, partial integration is a specific method within integration used when standard integration techniques are not applicable.
3. Integration relies on various rules, such as the power, chain, and substitution methods. In contrast, partial integration is based on the product rule for differentiation, allowing for a systematic approach to solving more complex integrals.

Integration vs Partial Integration

The difference between Integration and Partial Integration is that Integration is the simple anti-derivative of a function determined by using formulas. On the other hand, Partial Integration is a method used to partially break down and then integrate a rational fraction function with complex terms in the denominator following the LIATE rule.

Integration is the simplest form of an anti-derivation of a function. In other words, it is a mathematical method of uniting each part into a whole.

It calculates the bounded regions’ area or under the graphs’ curves. It has over twenty integration formulae for various functions such as trigonometry, algebra, inverse, and exponential.

Partial Integration is also called integration by parts. It is one of the methods of integration devised by mathematician Brook Taylor in 1715.

The partial integration formula thereby calculates integrals easier by simplifying the integration of functions into products. Moreover, it works well with integral expressions, which do not have direct integration formulae.

Comparison Table

Parameters of Comparison	Integration	Partial Integration
Definition	It is the anti-derivation of a function in math.	An integration method. It is also called integration by parts.
Formula	There are over twenty integration formulae for each function (trigonometry, algebra, inverse, exponential)	The uv integration formula: ∫ udv = uv – ∫ v du
Use	It determines the volume, area, and other dimensions of many things.	It simplifies the expression for easy integration.
Types	Definite and Indefinite integrals.	No types
Rules	Integration is the opposite of derivation.	LIATE- Logarthem, Inverse Trigonometric, Algebraic, Trigonometric, and Exponential functions.

What is Integration?

Integration is the primary method to be taught in calculus, preceded by differentiation. Both Isaac Newton and Gottfried Wilhelm Leibniz individually developed integration in the late 17th century.

According to this theory, the area under a curve is the sum of infinite rectangles of infinite width.

Furthermore, there are two types of integration in calculus: definite and indefinite. The definite integral is the area under the curve with two fixed upper and lower limits.

On the other hand, an indefinite integral is an area under the curve with no upper and lower limits.

Also, with a function derivation, one can determine the anti-derivation by using formulae and techniques; this method is called integration.

Additionally, specific rules must be followed to solve integration, such as sum & difference, power, constant multiplication, and reciprocal rules.

Integrals of some functions can be attained using four methods: integration by substitution, decomposition, partial integration, and integration by partial fractions.

∫ is the symbol that represents the integral of a function. For example, ∫ 1.dx = x + C means the integration of 1 (a constant) is equal to the sum of X and C (Constant). 

What is Partial Integration?

Two functions are to be solved using this method. It is also known as integration by parts. Partial integration is one of the methods of integration proposed by mathematician Brook Taylor in 1715.

It simplifies the integration of the product of functions into integrals for easy calculation. This technique is to calculate integral expressions with no direct integration formulae, such as inverse trigonometric and logarithmic functions.

Partial Integration is to find antiderivatives of functions that do not have exact solutions for, like in the case of polynomials, the trigonometric, exponential, and logarithmic functions. 

∫ udv = uv – ∫ v du is the integration of the uv formula employed to solve a function by partial integration. The two functions, u and v, are the integrals to be solved.

Additionally, LIATE – Logarithmic, Inverse Trigonometric, Algebraic, Trigonometric, and Exponential is an ordered set of functions to be followed for partial integration.

Accordingly, the first step is correctly identifying u and v functions based on the LIATE.

So in such a way, the integration of (product of First Function and Second Function) is equal to the Difference of { product of (First Function) and (Integration of Second Function)} and Integration of { product of (Differentiation of First Function and Integration of Second Function)}. 

Main Differences Between Integration and Partial Integration 

1. Integration is the primary method in calculus used to find the anti-derivative of functions. Whereas, Partial integration is one of the methods of integration. 
2. The integration method is done by jotting down formulae and solving them. Meanwhile, partial integration uses int ∫ udv=u v- ∫ int v du. 
3. Integration is formulated by Issac Newton and Gottfried Wilhelm Leibniz in the late 17th century. Meanwhile, Partial integration was developed by the Mathematician Brook Taylor in 1715. 
4. Integration of a function helps determine the area under a curve in the graph. On the other hand, partial integration helps in simplifying the expression for easy integration. 
5. Integration abides with the fundamental rules such as, the power rule, sum rule, and multiplication rule. However, partial integration obeys only one rule named LIATE (Logarithmic, Inverse Trigonometric, Algebraic, Trigonometric, and Exponential). 

References

1. https://www.emerald.com/insight/content/doi/10.1108/eb039143/full/html
2. https://heinonline.org/hol-cgi-bin/get_pdf.cgi?handle=hein.journals/taxlr47&section=33