The Substitution Rule
MAT1242: Calculus II
University of Eswatini.
The Substitution Rule, also known as the u-substitution method, is a powerful technique used in
calculus to simplify integrals. It allows us to transform complex integrals into simpler forms by making a
substitution. This method is particularly useful when dealing with functions that involve composition,
such as nested functions or trigonometric expressions.
Let’s dive into the details of the Substitution Rule, step by step:
1. Understanding the Concept
The Substitution Rule is based on the chain rule for differentiation. It states that if we have an integral of
the form:
where (f) and (g) are functions of (x), we can make a substitution to simplify the integral. The idea is to
introduce a new variable, denoted as (u), such that:
Then, we express (dx) in terms of (du):
This allows us to rewrite the integral as:
2. Step-by-Step Procedure
Let’s break down the steps for using the Substitution Rule:
Step 1: Choose a Suitable Substitution
1. Identify a part of the integrand that resembles the derivative of another function. This part will
become (u).
2. Compute (du) by differentiating (u) with respect to (x).
MAT1242: Calculus II
University of Eswatini.
The Substitution Rule, also known as the u-substitution method, is a powerful technique used in
calculus to simplify integrals. It allows us to transform complex integrals into simpler forms by making a
substitution. This method is particularly useful when dealing with functions that involve composition,
such as nested functions or trigonometric expressions.
Let’s dive into the details of the Substitution Rule, step by step:
1. Understanding the Concept
The Substitution Rule is based on the chain rule for differentiation. It states that if we have an integral of
the form:
where (f) and (g) are functions of (x), we can make a substitution to simplify the integral. The idea is to
introduce a new variable, denoted as (u), such that:
Then, we express (dx) in terms of (du):
This allows us to rewrite the integral as:
2. Step-by-Step Procedure
Let’s break down the steps for using the Substitution Rule:
Step 1: Choose a Suitable Substitution
1. Identify a part of the integrand that resembles the derivative of another function. This part will
become (u).
2. Compute (du) by differentiating (u) with respect to (x).
