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U Substitution: Examples & Concept

Instructor: Neelam Mehta

Neelam has taught variety of math and science subjects. She has masters' degrees in Chemical Engineering and Instructional Technology.

In this lesson, you will learn how to use the substitution technique for integration and also learn to recognize the types of problems with which you can use this method.

The Substitution Method for Integration

Integration by substitution is one of the many methods for evaluation of integrals in calculus. Other methods of integration include the use of integration formulas and tables, integration by parts, partial fractions method and trigonometric substitution.

In this lesson, you will learn to recognize when to use the u substitution technique for integration. You will also learn the steps for completing u substitution through examples.

Substitution Formula and a Step-by-Step Example

Have you ever come across a type of problem that you have never seen before? One way to approach the new problem is by turning it into a familiar one. That is the underlying idea behind u substitution, which is the process of solving an unknown integral by converting it into an integral that you know how to solve.

In mathematical form, the substitution formula for integration can be written as shown below. This formula is only true if the function u has continuous derivatives over an interval and if the function f can be integrated on the range of g(x).

Substitution Rule

According to this formula, we want to change an integral that is a function of the variable x to an integral that is a function of the variable u. We can make this change by completing following three steps:

  1. Substitute: Begin by changing the integral from a function of x to a function of u. To do this, you have to identify the function g(x) in the integral that you would replace with u. Let u equal to g(x). Differentiate u with respect to x and solve for g'(x)dx in terms of du. Now you are ready to make a complete substitution in the original integral. Substitute u in place of g(x) in the given integral. Also, substitute the expression you obtained in terms of du for g'(x)dx. Your new integral should be completely in terms of u and not in terms of x.
  2. Integrate: Evaluate the new integral with respect to u.
  3. Replace: Replace u with g(x) in the integral solution.

Let's apply above steps to evaluate the integral in following example:

U Substitution - Example 1

So did you notice something in the above example? I selected the function inside of the parenthesis as my choice for u. Typically, the functions inside of parentheses are good candidates for u substitution. Is this the only time you can use u substitution? No, there are plenty of other types of problems where this method is useful. Let's look at some other examples to understand the types of problems that can be solved using this technique.

Example: Integrating Functions under a Radical Sign

The functions under radical signs are also good choices for assignment as u. In the following example, I chose to substitute the function under the radical sign with u.

U Substitution - Example 2

Example: Integrating Trigonometric Functions

Try using u substitution, the next time you see an integration problem with a trigonometric function. Below are two examples demonstrating how to apply substitution when faced with trigonometric functions.

U Substitution - Examples 3 and 4

Expressions in Denominator and Exponents

There are two other instances when you can easily apply the u substitution technique. If you encounter an integration problem with Euler's number raised to an exponent with algebraic expression, then the expression can be substituted with u. For example, e to the (5x+8). In this type of problem, let u = 5x + 8.

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