Different Methods of Integration

In situations where we are unable to perform general addition operations on a large scale, integration is used to add values on a smaller scale. However, there are a variety of integration methods available in mathematics that can be used to integrate the functions. There are several different integration methods that can be used to find the integral of a function, which makes evaluating the original integral much easier than it would be otherwise.

For situations in mathematics where general addition operations are not possible, integration can be used to add values on a large scale. In mathematics, there are many different methods for integrating functions. Integral and differentiation are also pairs of inverse functions, similar to addition and subtraction and multiplication and division, among other things. Anti-differentiation or integration is the term used to describe the process of determining the derivative of a function that has been given.

Different methods of integration

Listed below is a list of Integration Methods:

1. Substitution as a method of integration. 

2. Parts as a method of integration. 

3. Integration by Partial Fraction. 

4. Integration of a specific fraction. 

5. Integration Using Trigonometric Identities. 

Integration by substitution

The substitution method of integration is also referred to as the u-substitution method of integration in some circles. This method allows us to change the variable of integration in order to make the function more simple. It is similar to the rule of the reverse chain. In the case of ∫g(f(x)), for example, integration is of the form ∫g(f(x)). Then, by assuming that f(x) = u, we can replace the function f(x) with another variable. We differentiate f(x) = u, which implies f'(x) dx = du = dx = du/h(u), where h(u) = f'(x) using the f(x) = u-substitution, which implies f'(x) dx = du/h(u), where h(u) = f'(x). Please keep in mind that if we change the variable of integration, then the variable must be changed throughout the entire integral as well. As a result, the following is the integration formula when using the substitution method:

                   ∫g(f(x)) dx = ∫g(u)/h(u) du

Integration by parts

One of the most important methods of integration is the method of integration by parts. In situations where the function to be integrated is written as a product of two or more functions, it is used. It is referred to as the product rule of integration and the uv method of integration, among other names. Suppose f(x) and g(x) are two functions, and the product of the two functions is to be integrated. The formula to integrate f(x).g(x)  using the by parts method:

∫f(x).g(x) dx = f(x)∫g(x) dx – ∫(f′(x) [∫g(x) dx)]dx+c.

The first function is denoted by f(x), and the second function is denoted by g(x).

Choosing the first function for integrating by parts is done on the basis of the sequence provided below. This method of integration is also referred to as the ILATE or LIATE method of integration, and it is abbreviated as follows:

I Inverse Trigonometric Function.

L stands for Logarithmic Function.

A In algebra, the letter A stands for Algebraic  Function.

T is an abbreviation for Trigonometric Function.

E is an abbreviation for Exponential Function.

Integration by partial fraction

The integration of rational functions is accomplished through the use of this method of integration. When a rational function has a denominator, it is possible to decompose it and convert it into a series of simpler multiple rational functions. Integration by partial fractions is one of the most widely used integration methods today. The following is the integration formula for rational functions of the form f(x)/g(x):

∫[f(x)/g(x)]dx = ∫[p(x)/q(x)]dx + ∫[r(x)/s(x)]dx

where

In addition, f(x)/g(x) = p(x)/q(x) + r(x)/s(x) and g(x) = q(x).s(x)

Different forms of rational functions are now decomposed using specific forms of partial fractions in order to make the calculation as easy and straightforward as possible. See our integration by partial fractions page for more information on the different forms of integration and how to make the functions simpler.

Important notes on methods of integration

• When integrating functions, the sum or difference of functions can be used to decompose the functions into smaller parts whose individual integrals are known.

• After determining the integral of the function, always include the constant of integration in the equation.

Conclusion

Various methods of solving complex and simple problems of integration in calculus are included in the category of Integration Methods. For situations in mathematics where general addition operations are not possible, integration can be used to add values on a large scale.

In mathematics, there are many different methods for integrating functions. Integral and differentiation are also pairs of inverse functions, similar to addition and subtraction and multiplication and division, among other things. 

In mathematics, there are many different methods for integrating functions. The substitution method of integration is also referred to as the u-substitution method of integration in some circles. This method allows us to change the variable of integration in order to make the function more simple.

One of the most important methods of integration is the method of integration by parts. In situations where the function to be integrated is written as a product of two or more functions, it is used. The integration of rational functions is accomplished through the use of this method of integration. When a rational function has a denominator, it is possible to decompose it and convert it into a series of simpler multiple rational functions.