Integrals of Trigonometric Functions

The process of finding a derivative, a derivative of a function, or a rate of change in mathematics. In contrast to the abstract nature of the underlying theory, the practical method of differentiation uses three basic derivations, and four operating rules, and knows how to operate a function, pure algebra. It can be executed by operation. 

However, an indefinite integral is a function that takes an indefinite integral of another function. It is finally represented as an integral symbol (∫), a function, and a derivative of the function. Indefinite integrals are an easier way to symbolize indefinite integrals.

List of some important indefinite integrals of trigonometric functions: 

Below is a list of some important formulas for the basic trigonometric indefinite integrals to keep in mind. 

• ∫ sin x dx = cos x + C 
• ∫ cos x dx = sin x + C 
• ∫sec2xdx = tan x + C 
• ∫ cosec2xdx = cot x + C 
• ∫ secxtan x dx = sec x + C 
• ∫ cosec x cot x dx = cosec x + C 
• ∫tanxdx = ln | seconds x | + c 
• ∫ cot xdx = ln | sin x | + c 
• ∫sec xdx = ln | seconds x + tan x | + c 
• ∫ Cosecant xdx  = ln | cosec x – cot x | + c

Where dx is the derivative of x, C is the integral constant, and ln is the logarithm of the function in the modulus (||). In general, the problem of indefinite integrals based on trigonometric functions is solved by the permutation method.

Therefore, the general form of permutation integration  is: 

 ∫ f (g (x)). g` (x) .dx = f (t) .dx 

 (where t = g (x)) 

 Normally, Die’s method permutation integration replaces a function whose derivative also exists in the integrand. Very useful if you want to. This simplifies the function and allows you to integrate the function using basic integral formulas. In the 

 analysis, the method of integration by substitution is also known as the “reverse chain rule” or “U substitution method”. If it is set in a special format, you can use this method to find the integer value.

Integration Formulae: 

You can use integrated expressions to integrate algebraic expressions, trigonometric ratios, inverse trigonometric functions, logarithmic functions, and exponential functions. Function integration gives the original function from which the derivative was obtained. These integrals are used to find the indefinite integral of the function. Differentiating the function f over the interval I yields the function family of I. If you know the value of the function of I, you can determine the function f. The reverse process of this differentiation is called integration.

The integral formula is generally expressed as the following 6 sets of formulas. Basically, integration is a way to bring parts together. Expressions include basic integrals, trigonometric ratio integrals, inverse trigonometric functions, product products, and some advanced integrals. Integration is the opposite of differentiation. Therefore, the basic integral formula is 

 ∫ f` (x). dx = f (x) + C.

Basic Integration Formulas: 

Using the essential theorems of integrals, there are generalized outcomes acquired which can be remembered as integration formulation in indefinite integration. 

• ∫ xn. dx = x (n + 1)/ (n + 1) + C
• ∫ 1.dx = x + C
• ∫ ex. dx = ex + C
• ∫1/x. dx = log|x| + C
• ∫ ax. dx = ax /loga+ C
• ∫ ex[f(x) + f`(x)].dx = ex.f(x) + C

Integration of Trigonometric functions: 

The process of finding an integral is integration. Here are some important integral formulas to keep in mind for quick and quick calculations: For trigonometric functions, we’ll simplify them and rewrite them as integrated functions. This is a list of trigonometric and inverse trigonometric functions. 

 ∫ cosx. dx = sinx + C 

 ∫sinx. dx = cosx + C 

 ∫sec2x.dx = tanx + C 

 ∫cosec2x.dx = cotx + C

 ∫secx. tanx.dx = secx + C

 ∫cosecx.cotx.dx = cosecx + C

 ∫tanx.dx = log | secx | + C

 ∫cotx.dx = log | sinx | ​​+ C

 ∫secx. dx = log | secx + tanx | + C

 ∫cosecx.dx = log | cosecx cotx | + c

Integration formula reverser trigonometric Function： 

 ∫1/√（1-x2.dx = sin1x + C 

 ∫/1（1 – x2.dx = cos1x + C 

 ∫1/（1 + x2.dx = tan1x + C 

 ∫1/（1 + x2.dx = cot1x + C 

 ∫1/x√（x2-1.dx = sec1x + C 

 ∫1/x√（x2-1.dx = cosec1 x + C

CONCLUSION:

Trigonometric integration involves basic simplification techniques. These techniques use various trigonometric formulas that can be written in alternative formats that are easy to integrate. There are 6 inverse trigonometric functions. However, the integral rules only list the three integrals that lead to the inverse trigonometric function. This is because it is a negative version of what the other three use. The only distinction is whether or not the integrand is positive. Given a negative integrand, instead of remembering three more formulas, factor -1 and evaluate the integral using one of the formulas already provided. To conclude this section, look at another formula. An integral that leads to an inverse tangent function.