Integration by Parts
MAT1242: Calculus II
University of Eswatini.
Integration by Parts is a powerful technique in calculus that allows us to evaluate certain types of
integrals by transforming them into simpler forms. It is particularly useful when dealing with products of
functions. In this guide, we’ll explore the concept of integration by parts, step-by-step procedures, and
provide multiple examples to help you prepare for your exams.
1. Understanding the Concept
Integration by parts is based on the product rule for differentiation. The formula for integration by parts
is:
where:
● (u) is a differentiable function of (x).
● (dv) is an integrable function of (x).
● (du) is the derivative of (u) with respect to (x).
● (v) is the antiderivative of (dv).
The goal is to choose (u) and (dv) in a way that simplifies the integral on the right-hand side.
2. Step-by-Step Procedure
Let’s break down the steps for using integration by parts:
Step 1: Choose (u) and (dv)
1. Identify (u) and (dv) in the given integral.
2. Typically, choose (u) such that its derivative (du) is simpler than the original function.
Step 2: Compute (du) and (v)
1. Differentiate (u) to find (du).
2. Integrate (dv) to find (v).
Step 3: Apply the Integration by Parts Formula
1. Use the integration by parts formula:
MAT1242: Calculus II
University of Eswatini.
Integration by Parts is a powerful technique in calculus that allows us to evaluate certain types of
integrals by transforming them into simpler forms. It is particularly useful when dealing with products of
functions. In this guide, we’ll explore the concept of integration by parts, step-by-step procedures, and
provide multiple examples to help you prepare for your exams.
1. Understanding the Concept
Integration by parts is based on the product rule for differentiation. The formula for integration by parts
is:
where:
● (u) is a differentiable function of (x).
● (dv) is an integrable function of (x).
● (du) is the derivative of (u) with respect to (x).
● (v) is the antiderivative of (dv).
The goal is to choose (u) and (dv) in a way that simplifies the integral on the right-hand side.
2. Step-by-Step Procedure
Let’s break down the steps for using integration by parts:
Step 1: Choose (u) and (dv)
1. Identify (u) and (dv) in the given integral.
2. Typically, choose (u) such that its derivative (du) is simpler than the original function.
Step 2: Compute (du) and (v)
1. Differentiate (u) to find (du).
2. Integrate (dv) to find (v).
Step 3: Apply the Integration by Parts Formula
1. Use the integration by parts formula:
