Integral Calculus Practise Flashcards

This set covers vocabulary and fundamental concepts of Integral Calculus based on Chapter 7 of the mathematics textbook, including anti-derivatives, standard integration methods, and the Fundamental Theorem of Calculus.

Anti-derivative (or Primitive)

A function $F$ whose derivative is equal to a given function $f$, such that $\frac{d}{dx}F(x) = f(x)$, $\forall x \in I$.

Integration

The process of finding anti-derivatives (or primitives) of a function, which is the inverse process of differentiation.

Indefinite Integral

The collection of all anti-derivatives of a function $f$, denoted by $\int f(x)\,dx = F(x) + C$, where $C$ is the constant of integration.

Integrand

The specific function $f(x)$ being integrated within the integral expression $\int f(x)\,dx$.

Variable of Integration

The variable $x$ in the expression $\int f(x)\,dx$, indicating with respect to which variable the integration is performed.

Constant of Integration

Any real number $C$, considered as a constant function, added to the anti-derivative to represent the family of functions sharing the same derivative.

Definite Integral

An integral denoted by $\int_a^b f(x)\,dx$ that has a unique value, representing the area bounded by a curve, the x-axis, and specific ordinates $x = a$ and $x = b$.

Fundamental Theorem of Calculus

The connection between the indefinite integral and the definite integral, stating that if $F$ is an anti-derivative of $f$, then $\int_a^b f(x)\,dx = F(b) - F(a)$.

Method of Inspection

A technique for finding an anti-derivative by intuitively searching for a function whose derivative is the given function.

Integration by Substitution

A method of integration where the independent variable $x$ is transformed into a new variable $t$ via a substitution $x = g(t)$ to reduce the integrand to a standard form.

Integration by Partial Fractions

The decomposition of a proper rational function into a sum of simpler rational functions to facilitate integration.

Proper Rational Function

A rational function $\frac{P(x)}{Q(x)}$ where the degree of the polynomial $P(x)$ is less than the degree of the polynomial $Q(x)$.

Improper Rational Function

A rational function $\frac{P(x)}{Q(x)}$ where the degree of $P(x)$ is greater than or equal to the degree of $Q(x)$, which can be reduced by long division.

Integration by Parts

A technique for integrating the product of two functions, defined as: $\int f(x)g(x)\,dx = f(x)\int g(x)\,dx - \int [f'(x)\int g(x)\,dx]\,dx$.

Area Function

A function defined as $A(x) = \int_a^x f(t)\,dt$, representing the area of the region bounded by a continuous curve $y = f(x)$ from a fixed point $a$ to a variable point $x$.

First Fundamental Theorem of Integral Calculus

States that for a continuous function $f$ on $[a, b]$, the derivative of its area function $A(x)$ is the function itself: $A'(x) = f(x)$.

Lower Limit

The value $a$ in the definite integral $\int_a^b f(x)\,dx$, marking the start of the interval of integration.

Upper Limit

The value $b$ in the definite integral $\int_a^b f(x)\,dx$, marking the end of the interval of integration.