Integration of Trigonometric Functions

What is the Integration of a Trigonometric Function?

The integration of trigonometric functions as the name suggests is the process of calculating the integration or antiderivative of trigonometric functions. This is the reverse process of differentiation of trigonometric functions.

What are Basic Trigonometric Functions?

The basic trigonometric functions are: 

• sine (sin), 
• cosine (cos), 
• tangent (tan), 
• cotangent (cot), 
• secant (sec), and 
• cosecant (csc).

How do you Integrate Sine (sin) and Cosine (cos) functions?

To integrate the sine and cosine functions, we can use the following formulas:

• ∫ sin(x) dx = -cos(x) + C
• ∫ cos(x) dx = sin(x) + C

Where C is the constant of integration.

What is the Integration of the Tangent (tan) Trigonometric Function?

The integral of the tangent function is given as follows:

∫ tan(x) dx = -ln|cos(x)| + C

Where,

• ln represents the natural logarithm, and 
• C is the constant of integration.

How to Find the Integral of the Secant (Sec) Trigonometric Function?

The integral of the secant function is given as:

∫ sec(x) dx = ln|sec(x) + tan(x)| + C

Where,

• ln represents the natural logarithm, and 
• C is the constant of integration.

What is the Integration of the Cotangent (cot) Trigonometric Function?

The integral of the cotangent function can be calcualted using the following formula:

∫ cot(x) dx = ln|sin(x)| + C

Where,

• ln represents the natural logarithm, and 
• C is the constant of integration.

How to Find the Integral of the Cosecant (cosec) Function?

The integral of the cosecant function is given as:

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

Where,

• ln represents the natural logarithm, and 
• C is the constant of integration.

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.