Standard Integrals

This article focuses on Integral Calculus and Standard Integrals Involving Algebraic Expressions. Integral calculus is the branch of mathematics that helps to find the antiderivatives – that are also known as integrals – of a function. It is a subfield of calculus. The process of finding integrals which is the reverse of derivatives is known as integration. A family of curves is represented by the integral of a given function.

An expression formed from performing mathematical operations on variables and constants is an algebraic expression. Polynomials are those algebraic expressions with at least one term with the non-negative integral exponent of a variable.

The concept of Integral Calculus

• Integral calculus is the branch of mathematics that helps to find the antiderivatives – that are also known as integrals – of a function. It is a subfield of calculus

• It deals with the study of integrals and their properties

• The fundamental theorem of calculus relates integral calculus and differential calculus

• The various applications of integral calculus involve – finding the area between two curves, the centre of mass, kinetic energy and surface area, work and volume, the average value of a given function, and correlating distance, velocity and acceleration

• Integral calculus is made up of definite and indefinite integrals

• Below are some formulas of integral calculus where C is an arbitrary constant

∫ dx = x + C

∫a dx = ax + C

∫(1/x) dx = ln |x| + C

∫ex dx = ex + C

∫ax dx = (ax / ln a) + C

∫ln x dx = x ln x – x + C

∫sin x dx = -cos x + C

∫cos x dx = sin x + C

∫tan x dx = ln |sec x| + C or C – ln |cos x|

∫cot x dx = ln |sin x| + C

∫sec x dx = ln |sec x + tan x| + C

∫cosec x dx = ln |cosec x – cot x| + C

∫sec2 x dx = tan x + C

∫sec x tan x dx = sec x + C

∫cosec2 x dx = -cot x + C

∫tan2 x dx = tan x – x + C

The Concept of Integration and Integrals

• The process of finding integrals which is the reverse of derivatives is known as integration

• Integration is also known as anti-differentiation

• A family of curves is represented by the integral of a given function

• Integrals are generalised based on the type of function and the domain of the performed integration

Types of Integrals and their Properties

• Integrals can be classified into two types – Definite and Indefinite Integral

• Definite integrals give the region’s area in the plane bounded by the graph of a given function between two points in the real line

• Indefinite integrals are those which can provide anti-derivatives of a given function

• We will be focusing on indefinite integrals (or anti-derivatives)

• An example of an indefinite integral is –

If f(x) = 2x, then ∫f(x) dx = ∫2x dx = x2 + C

Some properties of indefinite integrals are listed below.

• The process of integration and differentiation are the reverse of each other

• That is, 

(d/dx) ∫f(x) dx = f(x) and ∫f’(x) dx = f(x) + C

• Two indefinite integrals with the same derivatives lead to the same family of curves and are equivalent

• That is,

(d/dx) ∫f(x) dx = (d/dx) ∫g(x) dx

∫f(x) dx = ∫g(x) dx + C

∫f(x) dx and ∫g(x) dx are equivalent

∫[f(x) + g(x)] dx = ∫f(x) dx + ∫g(x) dx

For a real number, say k,

∫k f(x) dx = k ∫f(x) dx. A generalised formula for a finite number of functions with k real numbers is obtained from this equation.

Introduction to the Different Methods of Performing Integration

There are three main methods to perform an integration. They are –

• Integration by substitution: A given integral ∫ f(x) dx can be transformed into another form by changing the independent variable x to t by substituting x = g (t)

• Integration using partial fractions: Suppose we want to evaluate ∫ [P(x)/Q(x)] dx, where [P(x)/Q(x)] is a proper rational function

• We can write the integrand as a sum of simpler rational functions by partial fraction decomposition

• Integration by parts

Introduction to Standard Integrals Involving Algebraic Expressions

For a given constant C, and two functions, say, f(x) and g(x), there are 2 basic properties of integrals.

∫ c f(x) dx = c ∫ f(x) dx

And, 

∫[f(x) + g(x)] dx = ∫f(x) dx + ∫g(x) dx

These two properties form the Integration of Algebraic Functions or Standard Integrals Involving Algebraic Expressions.

Conclusion

Through this easy-to-understand and well-written article on Calculus – Integral Calculus – Standard Integrals Involving Algebraic Expressions – students have been made familiar with the concepts of integral calculus and integration, standard formulas for integral calculus, integrals and their types meaning and properties of indefinite integrals, different methods of performing integration, and standard integrals involving algebraic expressions.