Techniques of Integration in Economics: Substitution, Integration by Parts, and Partial Fractions

Introduction
In economic analysis, integration allows us to go from marginal to total quantities, determine areas under curves, and model consumer and producer surplus. But many economic functions are not easy to integrate directly. That’s why we use specialized techniques to simplify integrals and make them solvable.

This post introduces three key techniques of integration used in economics and applied mathematics:

1. Substitution
2. Integration by Parts
3. Partial Fractions

1. Substitution Method (Change of Variables)
This technique simplifies a complicated function by substituting a part of it with a single variable, often denoted $u$.

Formula:

If $u = g(x)$, then:
$\int f(g(x))g'(x) dx = \int f(u) du$

Example:

Let $MC(q) = 6(2q + 1)^4$. Then:
$\int MC(q) dq = \int 6(2q + 1)^4 dq$
Let $u = 2q + 1$, so $du = 2 dq$, thus $dq = \frac{du}{2}$

$\int 6(2q + 1)^4 dq = 6 \int u^4 \cdot \frac{1}{2} du = 3 \int u^4 du = 3 \cdot \frac{u^5}{5} + C = \frac{3}{5}(2q + 1)^5 + C$

Economic Relevance: Used when marginal cost/revenue involves power expressions or nested linear terms.

2. Integration by Parts
This technique comes from the product rule in differentiation and is useful when integrating the product of two different types of functions.

Formula:

$\int u \cdot dv = uv – \int v \cdot du$

Where:

• $u$ is chosen to simplify when differentiated
• $dv$ is easily integrable

Example:

$\int x e^x dx$
Let $u = x \Rightarrow du = dx$
Let $dv = e^x dx \Rightarrow v = e^x$

$\int x e^x dx = x e^x – \int e^x dx = x e^x – e^x + C = e^x(x – 1) + C$

Economic Relevance: Appears in utility models, discounted cash flow analysis, and exponential growth functions.

3. Integration Using Partial Fractions
Used when dealing with rational functions $\frac{P(x)}{Q(x)}$, especially when the degree of the numerator is less than the degree of the denominator.

Example:

$\int \frac{1}{x^2 – 1} dx$
Factor denominator: $x^2 – 1 = (x – 1)(x + 1)$

Decompose:
$\frac{1}{x^2 – 1} = \frac{A}{x – 1} + \frac{B}{x + 1}$
Solve for A and B (algebraically), then integrate:
$\int \left( \frac{1}{2(x – 1)} – \frac{1}{2(x + 1)} \right) dx = \frac{1}{2} \ln|x – 1| – \frac{1}{2} \ln|x + 1| + C$

Economic Relevance: Often used in microeconomic models involving elasticity, equilibrium, or rational price functions.

Tips for Choosing a Method

Conclusion
Mastering these integration techniques expands your ability to solve real-world economic problems involving cost, revenue, utility, and more. Whether you’re working with demand curves, marginal analysis, or complex revenue functions, substitution, integration by parts, and partial fractions turn difficult integrals into solvable ones.

With consistent practice, these methods become invaluable tools in your mathematical economics toolkit.

Further Reading

• Chiang & Wainwright, Fundamental Methods of Mathematical Economics
• Stewart, J. Calculus for the Life and Social Sciences