Methods of Integration

Before moving on to the integration method, let’s first recall the concept of integration. Integration is the process of combining very small strips of a figure to get the total area of the figure. Shows the area under the curve of the function. Use different integration methods to find the integrals of complex functions. To simplify the integration problem, you need to identify the type of function you want to integrate and then apply an integration method that facilitates the solution. We also use trigonometric expressions and identities as an integration method to simplify the trigonometric functions to be integrated.

List of Integration Methods

You can use the test method to solve the integral of the function. However, this may not work, and the function must first be simplified before evaluating the integral. To simplify these complex functions, the analysis uses the integral method. Below is a list of various integration methods that can help simplify integration issues. 

• Integration with parts 

• Partial fraction decomposition integration method 

• Integration by substitution method 

• Integration by disassembly 

• Reverse chain rule 

• Integration with trigonometric formulas

• Integration by Substitution

• Integration by Parts

• Integration Using Trigonometric Identities

Integration by Parts

Part integration is one of the most important methods of integration. Used when the function to be integrated is described as the product of two or more functions. This is also known as the product rule of integrals and the uv integral method. If f (x) and g (x) are two functions and you want to consolidate their products, the formula to consolidate f (x). g (x) with the parts method is: 

 ∫f (x). g (x) dx = f (x) ∫g (x) dx − ∫ (f ′ (x) [∫g (x) dx)] dx + C 

Where f (x) Is the first function and g (x) is the second function. For integration with parts, the first features are selected in the following order: This method, also commonly referred to as the ILATE or LIATE integral method, is abbreviated as follows:

• I = Inverse trigonometric function 

• L = logarithmic function 

• A = Algebraic function 

• T = trigonometric function 

• E = exponential function

Application-Suppose the integrand is the product of two functions, an exponential function and a logarithmic function. Compared to the ILATE preferred form, the logarithmic function is chosen as the first function and the exponential function can be used as the second function for ease of evaluation. Therefore, to solve ∫ (ex) log (4x2) dx, you can easily get the result by choosing log (4x2) as the first function and ex as the second function.

Method of Integration Using Partial Fractions

 This integral method is used to integrate rational functions. It is used to decompose the denominator of a rational function and transform it into simpler rational functions. Partial fraction decomposition integration is an important integration method. The formula that integrates rational functions of the form f (x) / g (x) is: 

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

 Where 

 f (x) / g (x) = p (x) / q (x) + r (x) / s (x) and 

 g (x) = q (x). s (x) 

Now, various forms of rational functions are decomposed using specific forms of partial fractions, making calculations simpler and easier. See the Partial Fraction Integration page for details on each shape and how to simplify the function.

Integration by Substitution Method of Integration

The permutation method is also commonly referred to as the integral permutation method. This way you can change the integral variables to simplify the function. This is similar to the reverse chain rule. For example, there is an integral of the form ∫g (f (x)) dx. You can then replace f (x) with another variable, assuming f (x) = u. Distinguish f (x) = u. This means f` (x) dx = du ⇒ dx = du / h (u). Where h (u) = f'(x), f (x) = substitution. Keep in mind that if you change the integral variable, you need to change it for the entire integral. Therefore, the integral formula by the permutation method is as follows. 

 ∫g (f (x)) dx = ∫g (u) / h (u) du

Important Note on How to Integrate 

• Functions to be integrated can be decomposed into sums or differences of functions for which individual integrals are known. 

• Whenever you find the integral of a function, be sure to add the constant of integration.

Reverse Chain Rule

The reverse chain rule is one of the simplest and most common methods of integration because it is the reverse process of the chain rule in differentiation. Here we identify the derivative of the function to be integrated. This integral method is used when the integral is in the form ∫g` (f (x)) f'(x) dx. In this case, the integral is given by

 ∫ g'(f (x)) f'(x) dx = g (f (x)) + C.

Conclusion

Integration is a method of adding large values that cannot perform common addition operations. However, there are several integration methods used in mathematics to integrate functions. There are various integration methods used to find the integral of a function that makes it easier to evaluate the original integral. We will explain in detail various integration methods such as integration by parts, integration by substitution, and integration by partial fractions.