Glossary

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 $\frac{1}{a}\mathrm{arctan}\left(\frac{u}{a}\right) + C$ .

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.

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.

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 $f'(x) = 3x^2$ .

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.

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.

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.

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.

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 $\frac{1}{2} \cdot \frac{1}{4}(2x+1)^4 + C$ .

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

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 $\frac{1}{a}\mathrm{arctan}\left(\frac{u}{a}\right) + C$ .

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

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.

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

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.

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 $f'(x) = 3x^2$ .

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

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

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.

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.

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.

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.

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:

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

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.