How to Integrate By Parts: Step-by-Step Guide

Integration by parts is a versatile and powerful technique in integral calculus, enabling the integration of products of functions that are otherwise difficult to integrate. Understanding and applying this method effectively requires practice, particularly in choosing \( u \) and \( dv \) appropriately.

Formula of Integration by Parts

The formula for integration by parts is derived from the product rule of differentiation (\(d(uv) = u dv + v du\)) and is given by:

\( \int u \, dv = uv – \int v \, du \)

where \( u \) and \( dv \) are functions of \( x \).

Choosing \( u \) and \( dv \)

The choice of which part of the integrand to set as \( u \) and which as \( dv \) is crucial. A common strategy is to use the LIATE rule (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential functions), which suggests a priority order for choosing \( u \).

Steps in Integration by Parts

1. Identify \( u \) and \( dv \): Split the integrand into two parts: \( u \) and \( dv \).
2. Differentiate \( u \): Differentiate \( u \) to find \( du \).
3. Integrate \( dv \): Integrate \( dv \) to find \( v \).
4. Apply the Formula: Substitute \( u \), \( du \), and \( v \) into the integration by parts formula.
5. Simplify: Simplify the expression and integrate the remaining integral if necessary.

Example:

Consider \(\int x \, e^x \, dx\).

• Choose \( u = x \) (Algebraic) and \( dv = e^x \, dx \) (Exponential).
• Differentiate \( u \) to get \( du = dx \).
• Integrate \( dv \) to get \( v = e^x \).
• Apply the formula: \( \int x \, e^x \, dx = x e^x – \int e^x \, dx \).
• Simplify: \( x e^x – e^x + C \).

Repeated Application

Sometimes, integration by parts must be applied more than once, especially if the resulting integral is still not straightforward. This is common with higher powers of \( x \).

Use in Definite Integrals

Integration by parts can also be applied to definite integrals. After applying the formula, evaluate the resulting expression at the upper and lower limits of integration.

Tabular Integration

For integrals requiring repeated applications, the tabular method, also known as the tabular integration by parts, can be a more efficient approach. This method organizes the differentiation of \( u \) and the integration of ( dv ) in a table format, streamlining the process.