Integration Formulas, Definitions, Formulas, Classification and its Application

Integral formulae can be used to combine algebraic expressions, trigonometric ratios, inverse trigonometric functions, logarithmic functions, and exponential functions. Integrating functions yields the fundamental functions for which the derivatives were generated. If Integrals confuses you then leave all your worries behind and join Safalta School Online to ace your preparation. 

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The ant derivative of a function is obtained using these integration formulae. When we distinguish a function f in an interval I, we gain a family of functions in I. If we are aware of the values of the functions in I, we can determine the function f.

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Differentiation has an opposite process called integration. Moving on, let's take a closer look at the integration formulae used in integration operations.

What are Integration Formulas? 

The following sets of formulae provide a general presentation of the integration formulas. Basic integration formulae, integration of trigonometric ratios, inverse trigonometric functions, the product of functions, and a more complex set of integration formulas are all included. Integration is essentially a method of joining the parts to create the whole. It is differentiation's opposite action. The fundamental integration equation is thus f'(x) dx = f(x) + C. The following integration formulae are generated using this.

 

Let's discuss the formulas in detail!

Basic Integrational Formula 

The list of basic integral formulas are

• ∫ 1 dx = x + C
• ∫ a dx = ax+ C
• ∫ xn dx = ((xn+1)/(n+1))+C ; n≠1
• ∫ sin x dx = – cos x + C
• ∫ cos x dx = sin x + C
• ∫ sec2x dx = tan x + C
• ∫ csc2x dx = -cot x + C
• ∫ sec x (tan x) dx = sec x + C
• ∫ csc x ( cot x) dx = – csc x + C
• ∫ (1/x) dx = ln |x| + C
• ∫ ex dx = ex+ C
• ∫ ax dx = (ax/ln a) + C ; a>0,  a≠1

Integration Formulas for Trigonometric Functions 

Integration is the method of discovering the integral. Here are a few crucial integration formulae to keep in mind for quick computations. We simplify and recast trigonometric functions as integrable functions when dealing with them. The trigonometric and inverse trigonometric functions are listed below.

• ∫ cos x dx = sin x + C
• ∫ sin x dx = -cos x + C
• ∫ sec2x dx = tan x + C
• ∫ cosec2x dx = -cot x + C
• ∫ sec x tan x dx = sec x + C
• ∫ cosec x cot x dx = -cosec x + C
• ∫ tan x dx = log |sec x| + C
• ∫ cot x dx = log |sin x| + C
• ∫ sec x dx = log |sec x + tan x| + C
• ∫ cosec x dx = log |cosec x - cot x| + C

Integration Formulas for Inverse Trigonometric Functions

Here are the integral formulae that result in or provide an inverse trigonometric function as the outcome.

• ∫1/√(1 - x2) dx = sin-1x + C
• ∫ 1/√(1 - x2) dx = -cos-1x + C
• ∫1/(1 + x2) dx = tan-1x + C
• ∫ 1/(1 + x2 ) dx = -cot-1x + C
• ∫ 1/x√(x2 - 1) dx = sec-1x + C
• ∫ 1/x√(x2 - 1) dx = -cosec-1 x + C

Advanced Integration Formulas 

The following are the advanced integration formulas:

• ∫1/(x2 - a2) dx = 1/2a log|(x - a)(x + a| + C
• ∫ 1/(a2 - x2) dx =1/2a log|(a + x)(a - x)| + C
• ∫1/(x2 + a2) dx = 1/a tan-1x/a + C
• ∫1/√(x2 - a2)dx = log |x +√(x2 - a2)| + C
• ∫ √(x2 - a2) dx = x/2 √(x2 - a2) -a2/2 log |x + √(x2 - a2)| + C
• ∫1/√(a2 - x2) dx = sin-1 x/a + C
• ∫√(a2 - x2) dx = x/2 √(a2 - x2) dx + a2/2 sin-1 x/a + C
• ∫1/√(x2 + a2 ) dx = log |x + √(x2 + a2)| + C
• ∫ √(x2 + a2 ) dx = x/2 √(x2 + a2 )+ a2/2 log |x + √(x2 + a2)| + C

Classification of Integral Formulas

The following functions are used to categorise the integral formulae that were just mentioned. 

• Rational functions
• Irrational functions
• Trigonometric functions
• Inverse trigonometric functions
• Hyperbolic functions
• Inverse hyperbolic functions
• Exponential functions
• Logarithmic functions
• Gaussian functions

Application of Integral Formulas 

In general, there are two types of integrals. They are either definite or indefinite integrals.

Definite Integration Formulas

These are integrations when the ultimate value of the integral is predetermined and the limitations have already been specified.

b∫ag(x)dx

= G(b) – G(a)

Indefinite Integration Formulas

These integrations have an unlimited ultimate value since they lack a predetermined limit value. This uses the integration constant C. g'(x) = g(x) + C

 

The area bounded by curves is approximated, average distance, velocity, and acceleration-oriented problems are evaluated, the average value of a function is determined, the volume and surface area of solids are approximated, the arc length is estimated, and the kinetic energy of a moving object using improper integrals is determined using the integration formulas that have been discussed thus far.

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