TRIGONOMETRY FUNCTIONS AND INTEGRATION

Trigonometric functions, sometimes known as Circular functions, can be simply described as functions of a triangle’s angle. This means that these trigonometric functions determine the relationship between the angles and the sides of a triangle. Trigonometric functions are real-valued functions in mathematics that link a right-angled triangle’s angle to the ratios of two side lengths. They’re commonly employed in all geometry-related disciplines. Sin, cos, tan, cot, sec, and cosec are the basic trigonometric functions.

In mathematics, integration is the process of finding a function g(x) whose derivative, Dg(x), is equal to a given function f. (x). This is represented by the integral symbol ”   ” as in f(x), which is commonly referred to as the function’s indefinite integral.

And when trigonometric functions and integration comes at the same time we have to integrate the trigonometric functions.

Representation

F(x) is the integration of a function f(x), and it is represented by:   f(x)dx = F(x) + C

The equation’s R.H.S. stands for integral f(x) with regard to x.

F(x) is called the anti-derivative or the primitive function

The integrand is defined as f(x).

The integrating agent is known as dx.

The arbitrary constant C is also known as the constant of integration.

x is the variable of integration.

Integration of some basic trigonometric functions: –

Here are the integration of some the some basic trigonometric functions that will be helpful in finding other integration:-

• ∫sin x dx = -cos x + C
• ∫cos x dx = sin x + C
• ∫tan x dx = ln|sec x| + C
• ∫sec x dx = ln|tan x + sec x| + C
• ∫cosec x dx = ln|cosec x – cot x| + C = ln|tan(x/2)| + C
• ∫cot x dx = ln|sin x| + C
• ∫sec^2x dx = tan x + C
• ∫cosec^2x dx = -cot x + C
• ∫sec x tan x dx = sec x + C
• ∫cosec x cot x dx = -cosec x + C
• ∫sin kx dx = -(cos kx/k) + C
• ∫cos kx dx = (sin kx/k) + C

DIFFERENT METHODS OF INTEGRATION: –

Different techniques of tackling complex and simple integration problems in calculus are included in methods of Integration. To use a given technique of integration, we must first determine the type of integral involved, and then solve it using the most appropriate method of integration. There are several methods of integration that we are using:-

Substitution method of Integration: –

By substituting the additional variables for independent variables in this method of integration by substitution, every given integral is turned into a simple form of integral. 

As a result, the General Form of substitutional integration is:

                       ∫ f(g(x)).g’(x).dx = f(t).dx                                                                         (where t = g(x))

When we make a substitution for a function whose derivative is also present in the integrand, the method of integration by substitution is usually quite useful. The function becomes simpler as a result, and the basic integration formulas can then be employed to integrate the function.

Integration by parts:-

Integration by parts necessitates a unique approach to function integration in which the integrand function is a multiple of two or more functions.

Consider the integrand function f(x) and g(x). Then integration by parts is

Integral of the product of two functions = (First function – Second function Integral) – Integral of [(differentiation of the first function) – Second function Integral]

When integrating by parts, the initial function is chosen according to the sequence listed below. This method of integration is also known as the ILATE method of integration, which stands for:

• I: Stand for Inverse Trigonometric Function.
• L: Stands for Logarithmic Function.
• A: Stands for Algebraic Function. 
• T: Stands for trigonometric function.
• E: Stand for Exponential Function.

Integration using trigonometric Identities: –

When integrating a function with any form of the Trigonometric integrand, we employ trigonometric identities to simplify the function so that it can be easily integrated.

The following are a few trigonometric identities-

• Sin2x=½(1-Cos2x)
• Cos2x=½(1+Cos2x)
• Sin^3x=¼(3sinx-sin3x)
• Cos^3x=¼(3cosx+cos3x)

Integration using some particular function: –

Integration of a certain function necessitates the use of some key integration equations that may be used to convert other functions into the standard form of the integrand. A direct type of integration method can readily find the integration of these common integrands.

CONCLUSION: –

The technique of integrating relatively small strips of a figure to achieve the total area of the figure is known as integration. It calculates the area under a function’s curve. To obtain the integral of complex functions, we use a variety of integration methods. To make integral issues easier to tackle, we must first choose the type of function to be integrated and then apply the integration procedure. To simplify the trigonometric functions under integration, we also use trigonometric formulae and identities as a method of integration.

When the function to be integrated is complex, we employ integration methods to simplify the function into simple forms whose integration is known. Integration and differentiation procedures are identical because integration is simply the reverse process of differentiation.