Some Standard Substitution

Standard substitution is very useful when we have to convert very complicated functions into a simple form. We can use some standard substitution in inverse trigonometric function, differentiation of function or sometimes even integration of function. Here we will be discussing the most common standard substitution method. If we find any of the expressions involved in function, then we should first do the right substitution and then proceed further. If no such expression is found, then we can do mathematical operations without substitution.

Some Standard Substitution

The following are some of the most useful substitutes for simplifying the provided mathematical expressions and making the differentiation and integration process easier.

Integration by Substitution

Integration by substitution is utilised when the integration of a given function cannot be accomplished directly because the algebraic function is not in the standard form. Furthermore, the provided function can be reduced to its standard form by substituting appropriately. 

Consider the indefinite integral,∫f(x).dx, of a function f(x) for evaluation. By replacing x with g(t) and substituting, this integral can be changed into another form, as follows:

Performing Substitutional Indefinite Integration

The instructions below will help you in completing this approach of integration by substitution.

Step 1: For the given function to be reduced, choose a new variable, t.

Step 2: Where f(x) is integrated with respect to x, find the value of dx for the given integral.

Step 3: Make the necessary changes to the function f(x), as well as the new value dx.

Step 4: Integrate the function that you got from the substitution.

Step 5: To get the final solution, replace the original variable x in the final answer.

Let us help you understand this in another way,

Numericals related to Some Standard Substitution

Substitute the value:

Substitute the value:

Conclusion :

Substitution is a technique for simplifying a system of equations by expressing one variable in terms of another and thereby eliminating one variable from the equation. Then, calculate this equation and replace it till you find the solution. Some standard substitution aims to simplify one of the equations by rewriting it in terms of a single variable. Furthermore, the crucial point to remember is that you are constantly substituting equal values.