Integrals by Partial Fractions – Rational Function Integration

This section presents integral formulas often encountered when using algebraic simplification and partial fractions. These help solve rational and trigonometric integrals involving polynomial and irreducible quadratic expressions.

Standard Partial Function Integrals

\[ \int \frac{dx}{x} = \ln |x| + C \]
\[ \int \frac{dx}{ax + b} = \frac{1}{a} \ln |ax + b| + C \]
\[ \int \frac{ax + b}{cx + d} \, dx = \frac{a}{c}x + \frac{bc - ad}{c^2} \ln |cx + d| + C \]
\[ \int \frac{dx}{x^2 + a^2} = \frac{1}{a} \arctan \frac{x}{a} + C \]
\[ \int \frac{dx}{x^2 - a^2} = \frac{1}{2a} \ln \left| \frac{x - a}{x + a} \right| + C \]
\[ \int \frac{dx}{a^2 - x^2} = \frac{1}{2a} \ln \left| \frac{a + x}{a - x} \right| + C \]
\[ \int \frac{dx}{(x + a)(x + b)} = \frac{1}{a - b} \ln \left| \frac{x + b}{x + a} \right| + C, \quad (a \neq b) \]
\[ \int \frac{x \, dx}{(x + a)(x + b)} = \frac{1}{a - b} \left( a \ln |x + a| - b \ln |x + b| \right) + C, \quad (a \neq b) \]
\[ \int \frac{x \, dx}{x^2 - a^2} = \frac{1}{2} \ln |x^2 - a^2| + C \]
\[ \int \frac{x \, dx}{x^2 + a^2} = \frac{1}{2} \ln |x^2 + a^2| + C \]
\[ \int \frac{dx}{(x^2 + a^2)^2} = \frac{1}{2a^2} \cdot \frac{x}{x^2 + a^2} + \frac{1}{2a^3} \arctan \frac{x}{a} + C \]
\[ \int \frac{x \, dx}{(x^2 + a^2)^2} = -\frac{1}{2} \cdot \frac{1}{x^2 + a^2} + C \]
\[ \int \frac{dx}{(x^2 + a^2)(x + b)} = \frac{1}{a^2 + b^2} \left( \ln \left| \frac{x + b}{\sqrt{x^2 + a^2}} \right| + \frac{b}{a} \arctan \frac{x}{a} \right) + C \]
\[ \int \frac{x \, dx}{(x^2 + a^2)(x + b)} = \frac{1}{a^2 + b^2} \left( \arctan \frac{x}{a} - b \ln \left| \frac{x + b}{\sqrt{x^2 + a^2}} \right| \right) + C \]

Terminology

Partial Fraction: The expression of a rational function as a sum of simpler fractions.
Rational Function: A ratio of two polynomials.
Irreducible Quadratic: A quadratic that cannot be factored over the real numbers (e.g., \( x^2 + a^2 \)).
Logarithmic Form: Integrals involving rational expressions often simplify to logarithms.
Inverse Trig Result: Some integrals involving quadratics lead to arctangent expressions.

Applications

Used to solve integrals involving rational expressions in algebra and calculus.
Important in engineering and physics to evaluate circuit models and dynamic systems.
Used in probability theory for normal distribution and density functions.
Critical in Laplace transforms and differential equation solutions.
Helps decompose complex fractions in real analysis and signal processing.

Integration - Integrals By Partial Functions

Integrals by Partial Functions

Special Integral Forms and Techniques

Standard Partial Function Integrals

Terminology

Applications

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