Some Important Integrals of Trigonometric Functions

Following is the list of some important formulae of indefinite integrals on basic trigonometric functions to be remembered as follows:

• ∫ sin x dx = -cos x + C
• ∫ cos x dx = sin x + C
• ∫ sec² x dx = tan x + C
• ∫ cosec² x dx = -cot x + C
• ∫ sec x tan x dx = sec x + C
• ∫ cosec x cot x dx = -cosec x + C
• ∫ tan x dx = ln | sec x | + C
• ∫ 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

Where dx is the derivative of x, C is the constant of integration and ln represents the logarithm of the function inside modulus (| |).

Generally, the problems of indefinite integrals based on trigonometric functions are solved by the substitution method. So let’s discuss more about the integration by substitution method as follows:

Integration is the process of summing up small values of a function in the region of limits. It is just the opposite to differentiation. Integration is also known as anti-derivative. We have explained the Integration of Trigonometric Functions in this article below.

Below is an example of the Integration of a given function.

e.g., Consider a function, f(y) = y². 

This function can be integrated as: 

∫y²dy = [Tex]\frac{y^{2+1}}{2+1}~+~C[/Tex]

However, an indefinite integral is a function that takes the anti-derivative of another function. It is represented as an integral symbol (∫), a function, and a derivative of the function at the end. The indefinite integral is an easier way to symbolize an anti-derivative.

Let’s learn what is integration mathematically, the integration of a function f(x) is given by F(x) and it is represented by:

∫f(x)dx = F(x) + C

Here R.H.S. of the equation means integral of f(x) with respect to x, F(x) is called anti-derivative or primitive, f(x) is called the integrand, dx is called the integrating agent, C is called constant of integration or arbitrary constant and x is the variable of integration.