A

Adjust and Compensate

Criticality: 3

A method used in integration by inspection where you multiply and divide by a constant to make the integrand match the exact derivative form required for the reverse chain rule.

Example:

If you need to integrate $\int x \sin(x^2) , dx$ , you can adjust and compensate by rewriting it as $\frac{1}{2} \int 2x \sin(x^2) , dx$ to match the derivative of $\cos(x^2)$ .

Antiderivative

Criticality: 3

A function $F(x)$ is an antiderivative of $f(x)$ if the derivative of $F(x)$ is $f(x)$, i.e., $F'(x) = f(x)$.

Example:

The function $x^3$ is an antiderivative of $3x^2$ , because the derivative of $x^3$ is $3x^2$ .

Arcsin

Criticality: 2

The inverse trigonometric function of sine, denoted as $\mathrm{arcsin}(x)$ or $\sin^{-1}(x)$, which returns the angle whose sine is $x$.

Example:

When solving integrals that resemble $\int \frac{1}{\sqrt{a^2 - u^2}} , du$ , the result will typically involve the arcsin function, like $\mathrm{arcsin}\left(\frac{u}{a}\right)$ .

Arctan

Criticality: 2

The inverse trigonometric function of tangent, denoted as $\mathrm{arctan}(x)$ or $\tan^{-1}(x)$, which returns the angle whose tangent is $x$.

Example:

Integrals of the form $\int \frac{1}{a^2 + u^2} , du$ are solved using the arctan function, yielding $\frac{1}{a}\mathrm{arctan}\left(\frac{u}{a}\right) + C$ .

C

Chain Rule

Criticality: 2

A fundamental differentiation rule used to find the derivative of a composite function, stating that $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$.

Example:

To find the derivative of $(3x+2)^5$ , you apply the chain rule, differentiating the outer power function and then multiplying by the derivative of the inner linear function.

Completing the Square

Criticality: 3

A powerful algebraic technique used to rewrite a quadratic expression into a perfect square trinomial plus a constant, typically in the form $(x+h)^2 + k$ or $a(x+h)^2 + k$.

Example:

To simplify the denominator of $\int \frac{1}{x^2 - 8x + 20} , dx$ , you would use completing the square to transform $x^2 - 8x + 20$ into $(x-4)^2 + 4$ .

Composite Function

Criticality: 2

A function formed by applying one function to the results of another function, denoted as $f(g(x))$.

Example:

In the expression $\sin(x^2)$ , $\sin(u)$ is the outer function and $u=x^2$ is the inner function, making $\sin(x^2)$ a composite function.

Composite Function

Criticality: 2

A function formed by combining two functions, where the output of one function becomes the input of the other, often written as $ f(g(x)) $. Identifying the inner function $ g(x) $ is key for u-substitution.

Example:

The function $e^{\cos(x)}$ is a composite function where $e^x$ is the outer function and $\cos(x)$ is the inner function.

Constant of Integration

Criticality: 2

An arbitrary constant, denoted by 'C', added to the result of an indefinite integral because the derivative of a constant is zero, meaning there are infinitely many antiderivatives.

Example:

When you integrate $\int 2x , dx$ , the answer is $x^2 + C$ , where $C$ is the constant of integration, representing any possible constant value.

Constant of Integration

Criticality: 2

An arbitrary constant, denoted by $ C $, that must be added to the result of every indefinite integral. This accounts for the fact that the derivative of any constant is zero.

Example:

When finding the indefinite integral of $x^2$ , the result is $\frac{1}{3}x^3 + C$ , where $C$ is the constant of integration.

Constant of Integration (C)

Criticality: 3

An arbitrary constant added to the antiderivative of a function to represent the family of all possible antiderivatives.

Example:

When integrating $f(x) = x$ , the result is $\frac{1}{2}x^2 + C$ , where C is the Constant of Integration.

D

Definite Integral

Criticality: 3

An integral with specified upper and lower limits of integration, which evaluates to a specific numerical value representing the net signed area under the curve between those limits.

Example:

The integral $\int_0^\pi \sin(x) , dx$ is a definite integral that calculates the area under the sine curve from 0 to $\pi$ .

Differentiate

Criticality: 3

In the context of u-substitution, this refers to the crucial step of finding the derivative of the chosen substitution $ u $ with respect to the original variable $ x $, typically expressed as $ \frac{du}{dx} $.

Example:

If we choose $u = x^2 + 5$ for a substitution, we must differentiate it to find $\frac{du}{dx} = 2x$ , which helps in replacing $dx$ .

Dividend

Criticality: 2

In a division operation, the dividend is the polynomial that is being divided.

Example:

In the expression $\frac{x^3 + 6x^2 - 9x - 11}{x-2}$ , $x^3 + 6x^2 - 9x - 11$ is the Dividend.

Divisor

Criticality: 2

In a division operation, the divisor is the polynomial by which the dividend is divided.

Example:

When simplifying $\frac{x^2 - 4}{x + 2}$ , $x + 2$ is the Divisor.

F

Function Raised to a Power

Criticality: 3

A specific pattern of integration where the integrand is of the form $f'(x) \cdot [f(x)]^n$, which integrates to $\frac{1}{n+1} [f(x)]^{n+1} + C$.

Example:

Integrating $\int 3x^2 (x^3 + 5)^4 , dx$ is an example of integrating a function raised to a power, where $f(x) = x^3 + 5$ and $f'(x) = 3x^2$ .

I

Indefinite Integral

Criticality: 3

The general form of the antiderivative of a function, representing all possible functions whose derivative is the given function.

Example:

Finding $\int (2x + 3) , dx = x^2 + 3x + C$ is calculating an Indefinite Integral.

Indefinite Integral

Criticality: 3

The collection of all antiderivatives of a given function $f(x)$, represented by $\int f(x) \, dx = F(x) + C$, where $F'(x) = f(x)$ and $C$ is an arbitrary constant of integration.

Example:

Finding the indefinite integral of $e^x$ gives $e^x + C$ , signifying that any function of the form $e^x$ plus a constant has $e^x$ as its derivative.

Indefinite Integral

Criticality: 3

An integral without specified limits, representing the general antiderivative of a function. Its result is a family of functions, differing only by a constant.

Example:

The expression $\int \frac{1}{x} , dx = \ln|x| + C$ is an indefinite integral, yielding a function plus a constant.

Integrate

Criticality: 3

The final step in solving an integral after substitution, where the simplified expression in terms of the new variable $ u $ is found by applying standard integration rules.

Example:

After substituting and simplifying an integral to $\int e^u , du$ , we then integrate it to get $e^u + C$ before substituting back the original variable.

Integrating Composite Functions

Criticality: 3

This process involves finding the antiderivative of a function that is composed of another function, often by recognizing and reversing the chain rule.

Example:

When you integrate $\int 2x \cos(x^2) , dx$ , you are integrating composite functions because $\cos(x^2)$ is a composite function and $2x$ is the derivative of its inner part.

Integrating f'(x)/f(x)

Criticality: 3

A special case of integration where the numerator is the derivative of the denominator, resulting in $\ln |f(x)| + C$.

Example:

When you encounter $\int \frac{2x}{x^2 + 1} , dx$ , you can quickly solve it by recognizing the integrating f'(x)/f(x) pattern, yielding $\ln|x^2+1| + C$ .

Integration Limits

Criticality: 3

The numerical values (upper and lower bounds) that define the interval over which a definite integral is evaluated. When using u-substitution for definite integrals, these limits must be transformed into the new $ u $ variable.

Example:

For $\int_1^3 (2x+1) , dx$ , 1 and 3 are the integration limits. If we substitute $u = 2x+1$ , these limits would change to $u=3$ and $u=7$ respectively.

Integration by Substitution (u-Substitution)

Criticality: 3

A powerful calculus technique used to simplify complex integrals by transforming them into an easier form through a change of variable, typically denoted by $ u $. It's particularly effective for integrals involving composite functions.

Example:

To solve $\int 2x \sin(x^2) , dx$ , we use integration by substitution by letting $u = x^2$ , which simplifies the integral to $\int \sin(u) , du$ .

Inverse Operations

Criticality: 1

Mathematical operations that undo each other, such as addition and subtraction, or differentiation and integration.

Example:

If you differentiate $x^3$ to get $3x^2$ , then integrating $3x^2$ brings you back to $x^3$ (plus a constant), illustrating that differentiation and integration are inverse operations.

M

Main Function

Criticality: 2

In the context of integrating composite functions, this refers to the outer function or the primary structure within the integrand that dictates the overall form of the antiderivative.

Example:

In $\int x(5x^2 - 2)^6 , dx$ , the $(5x^2 - 2)^6$ part is considered the main function because its structure guides the integration process.

P

Polynomial Long Division

Criticality: 3

A method used to divide polynomials, similar to numerical long division, to simplify rational functions or find factors.

Example:

When you divide $x^3 - 8$ by $x - 2$ to get $x^2 + 2x + 4$ , you are performing Polynomial Long Division.

Q

Quadratic Expression

Criticality: 3

A polynomial of degree two, generally written in the standard form $ax^2 + bx + c$, where $a$ is not equal to zero.

Example:

The function $y = 2x^2 + 3x - 5$ is a quadratic expression that graphs as a parabola and is often encountered in optimization or integration problems.

R

Rational Functions

Criticality: 2

A function that can be expressed as the ratio of two polynomials, where the denominator is not zero.

Example:

The function $f(x) = \frac{x^2 + 3x - 1}{x - 5}$ is a Rational Function.

Remainder

Criticality: 2

The leftover part of the dividend after a division operation, which cannot be further divided by the divisor to yield a polynomial term.

Example:

If you divide $x^2 + 5$ by $x + 1$ , the result is $x - 1$ with a Remainder of $6$ .

Reverse Chain Rule

Criticality: 3

A technique used in integration that involves working backward from the result of a differentiation using the chain rule to find the original function.

Example:

To integrate $\int (2x+1)^3 , dx$ , you might think of what function, when differentiated, would produce $(2x+1)^3$ , applying the reverse chain rule to get $\frac{1}{2} \cdot \frac{1}{4}(2x+1)^4 + C$ .

S

Standard Integrals

Criticality: 3

A set of fundamental integral forms whose antiderivatives are well-known and frequently applied in calculus, serving as building blocks for more complex integration problems.

Example:

Recognizing that $\int \frac{1}{\sqrt{1 - x^2}} , dx$ is a standard integral allows you to immediately write its solution as $\mathrm{arcsin}(x) + C$ .

Substitution

Criticality: 3

The core process within integration by substitution, where an expression in the original integral is replaced by a new variable (usually $ u $) to simplify the integrand. After integration, the original variable is substituted back.

Example:

In the integral $\int x \sqrt{x+1} , dx$ , the substitution $u = x+1$ allows us to rewrite $x$ as $u-1$ and $dx$ as $du$ , simplifying the expression.

Glossary

Adjust and Compensate

Antiderivative

Arcsin

Arctan

Chain Rule

Completing the Square

Composite Function

Composite Function

Constant of Integration

Constant of Integration

Constant of Integration (C)

Definite Integral

Differentiate

Dividend

Divisor

Function Raised to a Power

Indefinite Integral

Indefinite Integral

Indefinite Integral

Integrate

Integrating Composite Functions

Integrating f'(x)/f(x)

Integration Limits

Integration by Substitution (u-Substitution)

Inverse Operations

Main Function

Polynomial Long Division

Quadratic Expression

Rational Functions

Remainder

Reverse Chain Rule

Standard Integrals

Substitution