Integration Techniques Cheat Sheet

This cheat sheet summarizes key integration techniques including partial fractions, integration by parts, trigonometric substitutions, and combining methods. It provides formulas, steps, and examples for solving various types of integrals.

Core Principles

Integration is a skill, not an algorithm; persistence is key.
Partial fractions decompose rational functions into simpler, integrable forms.
Integration by parts reverses the product rule to solve integrals of products.
Trigonometric substitutions simplify integrals involving specific radical forms.
Combining techniques may be necessary for complex integrals.

Action Steps

Identify the type of integral (rational, product, trigonometric, etc.).
For rational functions, factor the denominator.
Decompose into partial fractions if the function is rational.
Use integration by parts for products of functions (e.g., polynomial times exponential/trigonometric).
Apply trigonometric substitution for integrals with $\sqrt{a^2-x^2}$ or $\sqrt{a^2+x^2}$.
If the integrand is improper (degree of numerator >= degree of denominator), use polynomial division first.
Combine techniques if a single method is insufficient.
Always check your answer by differentiating.

Formulas

$ \int \frac{A}{ax+b} dx = \frac{A}{a} \ln|ax+b| + C $
$ \int \frac{A}{(ax+b)^n} dx = \frac{A}{(n-1)(ax+b)^{n-1}} \times \frac{1}{a} + C \text{ for } n \neq 1 $
$ \int \frac{Dx+E}{ax^2+bx+c} dx \text{ (irreducible quadratic)} $
$ \int u dv = uv - \int v du $
$ \int f(x)g'(x) dx = f(x)g(x) - \int f'(x)g(x) dx $
$ \int_a^b f(x)g'(x) dx = [f(x)g(x)]_a^b - \int_a^b f'(x)g(x) dx $
$ \int \frac{1}{\sqrt{a^2-x^2}} dx = \arcsin(\frac{x}{a}) + C $
$ \int \frac{1}{a^2+x^2} dx = \frac{1}{a} \arctan(\frac{x}{a}) + C $

Key Terms

Rational Function: A function that can be written as the ratio of two polynomials, P(x)/Q(x).
Partial Fractions: Decomposing a rational function into a sum of simpler rational functions with linear or irreducible quadratic denominators.
Integration by Parts: A technique used to integrate the product of two functions, derived from the product rule for differentiation.
Trigonometric Substitution: Replacing a variable with a trigonometric function to simplify integrals involving specific radical expressions.
Improper Rational Function: A rational function where the degree of the numerator is greater than or equal to the degree of the denominator.
Irreducible Quadratic Factor: A quadratic expression that cannot be factored into real linear factors (e.g., $x^2+1$).

Pro Tips

For integration by parts, choose 'u' such that its derivative simplifies, and 'dv' such that it's easily integrable.
If an integration by parts step improves the integrand, continue applying the technique.
Remember the derivative of arctan(x) is $1/(1+x^2)$ and arcsin(x) is $1/\sqrt{1-x^2}$.
For definite integrals, evaluate the antiderivative at the limits and subtract.
Polynomial division is crucial for improper rational functions before partial fraction decomposition.

Pitfalls to Avoid

Forgetting the '+ C' in indefinite integrals.
Incorrectly factoring the denominator in partial fractions.
Errors in algebraic manipulation during integration by parts.
Confusing the roles of f(x) and g'(x) in integration by parts.
Not simplifying or checking the final answer.

Real World Examples

Calculating the area under a curve defined by a complex rational function.: Partial fractions simplify the integrand for easier integration.
Finding the work done by a variable force or the volume of a solid of revolution.: Integration by parts is often used in physics and engineering problems.
Problems involving circular or hyperbolic motion.: Trigonometric and hyperbolic substitutions are common.

People

Isaac Newton: Pioneered calculus, including integration concepts.
Gottfried Wilhelm Leibniz: Co-inventor of calculus, developed much of the notation used today.