Integral as an anti-derivative

Integration of Functions:

In mathematics, the antiderivative of an integrable function is a primitive of it in one variable defined as the indefinite integral concerning that variable. An antiderivative of a continuous function is called an indefinite integral and is usually denoted by ∫f(x)dx. The integration of functions is the opposite of differentiation as in differentiation, we divide the particles into small parts and on standard integrals, we just combine all the divided parts to get the function back. 

For example, the integration of x will be ½ x^2+ c where c is taken as the constant. The constant is used as when the function is reversed, there is no proper idea about the last term that will be included while integrating it. Therefore, a “c” is used to show that a constant will appear if we integrate the function there. 

Standard Integrals :

The list of basic integrals consists of the rules that should be well known by all the students before moving to Integration of Trigonometric Functions. These formulas pave the way for a better understanding of the topic and help the students to understand the topic better.

The principal values of the function of a complex variable may be considered as an analytic function in general. If the characteristics of a non-analytic but continuous function are C(z) on the real axis, then its principal value is defined by

1. ∫ x^n dx = x^n+1/(n+1) + C 
2. ∫ x dx = ln |x| + C 
3. ∫ e^x dx = e^x + C

The Cauchy principal value of the integral around the singularity is −iπ if the integration is done in the complex plane. On the real line it will use a completely different value of C. With these standard integrals, the students will be able to solve several problems given to them. 

Integration of Trigonometric Functions :

The integration of trigonometric functions includes the basic principles that have been involved in the previous formulas. The Fundamental Integrals involving algebraic trigonometric involves the algebraic functions along with trigonometric functions in the problems. The formulas given will be the anti-derivatives of the trigonometric functions in it.

1. ∫sin x dx = − cos x + C 
2. ∫ cos x dx = sin x + C 
3. ∫ tan x dx = − ln (cos x)   + C 
4. ∫ cot x dx = ln  |sin x|  + C
5. ∫sec^2x dx = tan x + C 
6. ∫cosec^2x dx = − cot x + C
7. ∫sec x dx = ln (sec x + tan x) + C 
8. ∫cosec x dx = -ln |cosec x − cot x|+ C 
9. ∫sinh x dx = cosh x + C 
10. ∫cosh x dx = sinh x + C 
11. ∫tanh x dx = ln |cosh x| + C

Inverse Trigonometric Functions :

The inverse trigonometric integration of functions is used in the higher-order problems where algebraic trigonometry will be there. There are a total of 6 formulas that will come into the play.

1. ∫ 1/√(a^2-u^2) du = sin^-1(u/a) + c
2. ∫1/(a^2+u^2)du = 1/a tan^-1(u/a) +c
3. ∫1/(u√(u^2-a^2)) du = 1/a sec^-1(u/a) + c
4. ∫sin^-1 u du = u sin^-1u + √(1-u^2)+ c
5. ∫tan^-1 u du = u tan^-1u – ½ ln(1+u^2) + c
6. ∫cos^-1u du = u cos^-1u – √(1-u^2)+ c

These inverse trigonometric integration of functions will help the students to understand the concepts clearly and do the problems easily that will be based on it.

Hyperbolic Trigonometric Functions

The Hyperbolic Trigonometric Integration of Functions is used where the domains are not restricted and we need to be careful while dealing with the problems that will be associated with it. The formulas derived from here will be used in the problems that will appear completing the inverse trigonometric functions.

1. ∫Sinh u du = cosh u + c
2. ∫cosh u du = sinh u + c
3. ∫tanh u du = ln(cosh u) + c
4. ∫Sech u tanh u du = -sech u + c
5. ∫Cosech u coth u du = – cosech u + c
6. ∫Sech u du =  tan^-1 |sinhu| + c
7. ∫sech^2 u du =  tanh u + c
8. ∫cosech^2  u du = – coth u + c

Properties of Integration :

In this section, we will be looking at the 8 properties of Integration that helps the students to understand and get the problems done easily. These properties make the difficult questions way easier by transforming them into simpler equations.

Conclusion :

In this material we got to learn about the standard integrals and how Fundamental Integrals involving algebraic trigonometry are used. Apart from it, the use of integrals in the trigonometric functions which consists of inverse trigonometric functions and hyperbolic trigonometric functions. With these functions, the problems will get easier and students will face less difficulty while solving the problems. Besides this, the properties of definite integrals have also been shared here. These properties make sure that the problems of Integrations get way easier. The Integration of Trigonometric Functions can be learned easily through these properties.