Glossary

Antidifferentiation

Criticality: 3

The process of finding the antiderivative or indefinite integral of a function. It is the reverse operation of differentiation.

Example:

To find the position function from a velocity function, you would perform antidifferentiation.

Completing the Square for Integration

Criticality: 2

An algebraic technique used to transform quadratic expressions within an integrand into a perfect square trinomial, often to facilitate integration using inverse trigonometric forms or u-substitution.

Example:

To integrate $\int \frac{1}{x^2+6x+10} dx$ , you can use Completing the Square for Integration to rewrite the denominator as $(x+3)^2+1$ , which then leads to an arctan integral.

Constant of Integration (C)

Criticality: 3

An arbitrary constant added to the antiderivative of a function because the derivative of any constant is zero, meaning there are infinitely many antiderivatives for a given function.

Example:

After integrating $2x$ , we must add the Constant of Integration (C), resulting in $x^2 + C$ , to represent all possible antiderivatives.

Exponential and Logarithmic Integrals

Criticality: 3

Antiderivatives involving exponential functions ($e^x$) and natural logarithmic functions ($\ln|x|$), which have straightforward integration rules.

Example:

The integral of $e^x$ is simply $e^x + C$ , and the integral of $\frac{1}{x}$ is $\ln|x| + C$ , both being fundamental Exponential and Logarithmic Integrals.

Inverse Trigonometric Integrals

Criticality: 2

Antiderivatives that result in inverse trigonometric functions (arcsin, arccos, arctan, etc.), often arising from integrands involving specific quadratic forms.

Example:

The integral $\int \frac{1}{\sqrt{1-x^2}} dx$ is a classic Inverse Trigonometric Integral that evaluates to $\arcsin(x) + C$ .

Long Division for Integration

Criticality: 2

A technique used to simplify rational functions where the degree of the numerator is greater than or equal to the degree of the denominator, by dividing the polynomials before integrating.

Example:

Before integrating $\int \frac{x^3+x}{x-1} dx$ , you should perform Long Division for Integration to rewrite the expression into a simpler polynomial plus a remainder term.

Power Rule for Antiderivatives

Criticality: 3

A fundamental rule for integrating functions of the form x^n, where you add 1 to the exponent and divide by the new exponent, plus a constant of integration.

Example:

When integrating $x^5$ , apply the Power Rule for Antiderivatives to get $\frac{x^6}{6} + C$ .

Trigonometric Integrals

Criticality: 2

Antiderivatives of common trigonometric functions like sine, cosine, tangent, cotangent, secant, and cosecant, which often require memorization of their basic forms.

Example:

Knowing that the Trigonometric Integral of $\cos(x)$ is $\sin(x) + C$ is essential for solving many calculus problems.

U-substitution (U-sub)

Criticality: 3

A powerful technique used to simplify complex integrals, especially those involving composite functions, by replacing a part of the integrand with a new variable 'u'.

Example:

To integrate $\int x \sin(x^2) dx$ , you can use U-substitution by letting $u = x^2$ , which simplifies the integral significantly.