Glossary

Absolute Value (|x|)

Criticality: 2

The absolute value of a number $ x $, denoted as $ |x| $, is its non-negative value regardless of its sign. It ensures that the argument of a logarithm is always positive.

Example:

In $\int \frac{1}{x} , dx = \ln |x| + C$ , the absolute value ensures the logarithm is defined for both positive and negative $x$ .

Antiderivative

Criticality: 3

A function whose derivative is the original given function. It is the reverse operation of differentiation.

Example:

$F(x) = x^3$ is an antiderivative of $f(x) = 3x^2$ , because the derivative of $x^3$ is $3x^2$ .

Antiderivative

Criticality: 3

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

Example:

If the velocity of a particle is $v(t) = 3t^2$ , then its position function $s(t) = t^3 + C$ is an antiderivative of the velocity function.

Antiderivatives

Criticality: 3

Antiderivatives are functions whose derivative is the original function. They are also known as indefinite integrals and represent the reverse process of differentiation.

Example:

If the velocity of a particle is $v(t) = 3t^2$ , then its position function, an antiderivative, could be $s(t) = t^3 + C$ .

Constant of Integration ($C$)

Criticality: 3

An arbitrary constant added to the antiderivative of a function when computing an indefinite integral, representing the family of all possible antiderivatives.

Example:

When you integrate $2x$ , you get $x^2 + C$ . If you know the antiderivative passes through $(0, 5)$ , then $0^2 + C = 5$ , so $C=5$ , giving the specific antiderivative $x^2 + 5.

Constant of Integration (C)

Criticality: 3

An arbitrary constant added to the antiderivative of a function, representing the vertical shift of the family of antiderivatives.

Example:

When solving $\int \cos(x) , dx$ , you must include the constant of integration (C) to get $\sin(x) + C$ , acknowledging that $\sin(x)+1$ , $\sin(x)-7$ , etc., all have the same derivative.

Constant of Integration (C)

Criticality: 3

The constant of integration is an arbitrary constant added to the result of an indefinite integral. It accounts for the fact that the derivative of any constant is zero.

Example:

When integrating $\int 2x , dx$ , the result is $x^2 + C$ , where C can be any real number, like $x^2 + 5$ or $x^2 - 10$ .

Constant of Integration (C)

Criticality: 3

An arbitrary constant added to the indefinite integral of a function, accounting for the loss of information about constant terms during differentiation.

Example:

After integrating 3x², the result is x³ + C, because the derivative of any constant is zero, so we must include this unknown constant.

Derivatives

Criticality: 3

Derivatives represent the instantaneous rate of change of a function with respect to one of its variables. They are fundamental to understanding slopes of tangents and velocities.

Example:

If a car's position is given by $s(t) = t^2$ , its velocity, the derivative $s'(t) = 2t$ , tells you how fast it's moving at any time $t$ .

Differentiation

Criticality: 3

Differentiation is the mathematical process of finding the derivative of a function. It determines the rate at which a function's value changes.

Example:

Differentiation of $f(x) = x^3$ yields $f'(x) = 3x^2$ , showing the slope of the tangent at any point.

Differentiation (as inverse of integration)

Criticality: 2

Differentiation is the process of finding the derivative of a function, serving as the inverse operation to integration.

Example:

If you integrate $2x$ to get $x^2 + C$ , you can check your answer by performing differentiation on $x^2 + C$ , which correctly yields $2x$ .

Exponent

Criticality: 1

The power to which a number or expression is raised, indicating how many times the base is multiplied by itself.

Example:

In the term x⁷, the number 7 is the exponent, which is critical for applying the power rule of integration (add 1 to the exponent and divide by the new exponent).

Exponential Integral

Criticality: 3

An exponential integral refers to the integration of exponential functions, such as $ e^x $ or $ a^x $. The integral of $ e^{kx} $ is $ \frac{1}{k} e^{kx} + C $.

Example:

To model population growth, you might integrate a rate function like $\int 2e^{0.05t} , dt$ , which is an exponential integral.

Family of Antiderivatives

Criticality: 2

The set of all possible antiderivatives for a given function, which differ from each other only by a constant value.

Example:

For the function $f(x) = 4x^3$ , the family of antiderivatives is $x^4 + C$ , where $C$ can be any real number, like $x^4+1$ , $x^4-10$ , etc.

Indefinite Integral

Criticality: 3

An integral without upper and lower limits, representing the general antiderivative of a function, including the constant of integration.

Example:

The indefinite integral of $3x^2$ is $x^3 + C$ , representing all functions whose derivative is $3x^2$ .

Indefinite Integral

Criticality: 3

The indefinite integral of a function $f(x)$ is the antiderivative of $f(x)$, denoted by $\int f(x) \, dx$, representing a family of functions whose derivative is $f(x)$.

Example:

When you find the indefinite integral of $2x$ , you get $x^2 + C$ , representing all functions like $x^2$ , $x^2+5$ , etc., whose derivative is $2x$ .

Indefinite Integral

Criticality: 3

An indefinite integral is an integral without specific limits of integration, representing a family of functions whose derivative is the integrand. It always includes a constant of integration.

Example:

The indefinite integral of $2x$ is $x^2 + C$ , meaning any function of the form $x^2$ plus a constant will differentiate to $2x$ .

Indefinite Integral

Criticality: 3

The anti-derivative of a function, representing a family of functions whose derivative is the original function, differing only by a constant.

Example:

When you find the indefinite integral of 2x, you get x² + C, which represents all functions like x²+1, x²-5, etc., whose derivative is 2x.

Integral

Criticality: 3

The anti-derivative of a function, representing the area under the curve.

Example:

Calculating the integral of a velocity function over time gives you the total displacement.

Integrate

Criticality: 3

To integrate a function means to find its indefinite integral or antiderivative.

Example:

To find the total displacement from a velocity function, you would integrate the velocity function with respect to time.

Integrating Constant Multiples

Criticality: 3

A rule allowing a constant factor to be moved outside the integral sign when integrating a function multiplied by that constant.

Example:

When evaluating the integral of 5e^x, you can pull the 5 out front, making it 5 times the integral of e^x, which is an application of integrating constant multiples.

Integrating Sums and Differences

Criticality: 3

A fundamental rule stating that the integral of a sum or difference of functions is equal to the sum or difference of their individual integrals.

Example:

To find the integral of (x³ + cos x), you can apply the rule for integrating sums and differences by integrating x³ and cos x separately, then adding their results.

Integration

Criticality: 3

The mathematical process of finding the antiderivative of a function. It is the reverse operation of differentiation.

Example:

To find the total distance traveled given a velocity function, you would perform integration of the velocity function over time.

Integration

Criticality: 3

Integration is the mathematical process of finding the antiderivative of a function. It is used to calculate areas, volumes, and total changes from rates of change.

Example:

To find the total distance traveled given a velocity function, you would perform integration over a specific time interval.

Inverse Operations

Criticality: 3

Inverse operations are mathematical processes that reverse the effect of each other. In calculus, differentiation and integration are inverse operations.

Example:

Adding 5 and subtracting 5 are inverse operations; similarly, taking a derivative and then an indefinite integral (or vice versa) brings you back to the original function (plus a constant).

Inverse Trigonometric Integrals

Criticality: 3

Inverse trigonometric integrals are those that result in inverse trigonometric functions (like arcsin, arctan) as their antiderivatives. They typically involve specific fractional forms.

Example:

The integral $\int \frac{1}{1+x^2} , dx = \arctan x + C$ is a common inverse trigonometric integral.

Logarithm (ln x)

Criticality: 2

The natural logarithm, denoted as $ \ln x $, is the inverse function of the exponential function $ e^x $. It is crucial for integrating functions of the form $ 1/x $.

Example:

The integral of $\frac{1}{x}$ is $\ln |x| + C$ , highlighting the importance of the logarithm in calculus.

Power Rule for Integration

Criticality: 3

The power rule for integration states that $ \int x^n \, dx = \frac{1}{n+1} x^{n+1} + C $ for any real number $ n eq -1 $. It is used to integrate polynomial terms.

Example:

Using the power rule for integration, $\int x^5 , dx = \frac{1}{6} x^6 + C$ .

Reciprocal Trigonometric Integrals

Criticality: 2

Reciprocal trigonometric integrals involve finding the antiderivatives of functions like secant, cosecant, and cotangent, or their products. They are essential for a complete understanding of trigonometric calculus.

Example:

An example of a reciprocal trigonometric integral is $\int \sec^2 x , dx = an x + C$ .

Simplifying Expressions

Criticality: 2

The process of algebraically manipulating an integrand (e.g., expanding brackets, rearranging fractions) to make it easier to integrate using standard rules.

Example:

Before integrating (x+3)², you would first expand it to x² + 6x + 9, which is a crucial step of simplifying expressions for integration.

Specific Antiderivative

Criticality: 3

A unique antiderivative determined by using additional given conditions (like a point the function passes through) to find the exact value of the constant of integration.

Example:

If the general antiderivative is $x^2 + 3x + C$ and it passes through $(1, 2)$ , solving for $C$ yields $C=-2$ , resulting in the specific antiderivative $x^2 + 3x - [object Object]$ .

Trigonometric Integrals

Criticality: 3

Trigonometric integrals involve finding the antiderivatives of trigonometric functions like sine, cosine, and tangent. These are derived directly from their differentiation rules.

Example:

Calculating the area under a wave function often requires solving trigonometric integrals like $\int \cos(2x) , dx$ .