Integral Calculus Practise Flashcards

Introduction to Integral Calculus

Differential Calculus is primarily concerned with the concept of the derivative, which originated from the problem of defining tangent lines to function graphs and calculating their slopes. In contrast, Integral Calculus is motivated by the problem of defining and calculating the area of regions bounded by the graphs of functions. These two problems lead to the two forms of integrals: indefinite and definite integrals. Together, they constitute the field of Integral Calculus. Historically, the development of this field arose from efforts to solve two specific types of problems: finding a function when its derivative is known, and finding the area bounded by a function's graph under certain conditions.

A significant connection exists between indefinite and definite integrals known as the Fundamental Theorem of Calculus. This theorem transforms the definite integral into a practical tool for science and engineering, as well as for disciplines such as economics, finance, and probability. The mathematician G.W. Leibnitz (1646–1716) is a central figure in the history of these developments. James B. Bristol provides a thematic metaphor for the study of the subject, stating that just as a mountaineer climbs a mountain because it is there, a good mathematics student studies new material because it is there.

Integration as an Inverse Process of Differentiation

Integration is defined as the inverse process of differentiation. Rather than differentiating a function, we are provided with the derivative and tasked with finding the original function, referred to as the primitive or anti-derivative. If a function $f$ is differentiable in an interval $I$ such that its derivative $f'$ exists at each point, the natural question is whether we can determine the original function. The process of finding these anti-derivatives is called integration.

For example, we know that $\frac{d}{dx}(\sin(x)) = \cos(x)$, $\frac{d}{dx}\left(\frac{x^3}{3}\right) = x^2$, and $\frac{d}{dx}(e^x) = e^x$. Consequently, $\sin(x)$ is an anti-derivative of $\cos(x)$, $\frac{x^3}{3}$ is an anti-derivative of $x^2$, and $e^x$ is an anti-derivative of $e^x$. However, these anti-derivatives are not unique. Because the derivative of any constant function $C$ is zero, we can observe that $\frac{d}{dx}(\sin(x) + C) = \cos(x)$. Thus, there are infinitely many anti-derivatives for a given function, obtained by choosing $C$ as any arbitrary real number. In this context, $C$ is called the constant of integration or an arbitrary constant, acting as a parameter that generates a family of curves.

Geometrically, if there exists a function $F$ such that $\frac{d}{dx}(F(x)) = f(x)$ for all $x \in I$, then for any arbitrary real number $C$, $\frac{d}{dx}[F(x) + C] = f(x)$. The set $\{F + C, C \in \mathbb{R}\}$ denotes the entire family of anti-derivatives of $f$. It is a fundamental remark in calculus that functions possessing the same derivative differ only by a constant. If two functions $g$ and $h$ have the same derivatives such that $g'(x) = h'(x)$, then the derivative of their difference $(g - h)' = 0$, implying $g(x) - h(x) = C$.

Symbols, Terms, and Phrases in Integration

The mathematical notation for the indefinite integral of $f$ with respect to $x$ is given by $\int f(x) \, dx$. If $\frac{dy}{dx} = f(x)$, we write $y = \int f(x) \, dx = F(x) + C$. The following table summarizes the standard terminology:

1. $\int f(x) \, dx$: Integral of $f$ with respect to $x$.
2. $f(x)$ in $\int f(x) \, dx$: The Integrand.
3. $x$ in $\int f(x) \, dx$: The Variable of integration.
4. Integrate: The act of finding the integral.
5. An integral of $f$: A function $F$ such that $F'(x) = f(x)$.
6. Integration: The process of finding the integral.
7. Constant of Integration: Any real number $C$ treated as a constant function.

Standard Formulae for Derivatives and Integrals

Based on established derivative formulae, the following standard integral results are derived:

(i) $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$, where $n \neq -1$. Notably, $\int 1 \, dx = \int dx = x + C$. (ii) $\int \cos(x) \, dx = \sin(x) + C$ (iii) $\int \sin(x) \, dx = -\cos(x) + C$ (iv) $\int \sec^2(x) \, dx = \tan(x) + C$ (v) $\int \text{cosec}^2(x) \, dx = -\cot(x) + C$ (vi) $\int \sec(x)\tan(x) \, dx = \sec(x) + C$ (vii) $\int \text{cosec}(x)\cot(x) \, dx = -\text{cosec}(x) + C$ (viii) $\int \frac{dx}{\sqrt{1 - x^2}} = \sin^{-1}(x) + C$ (ix) $\int \frac{-dx}{\sqrt{1 - x^2}} = \cos^{-1}(x) + C$ (x) $\int \frac{dx}{1 + x^2} = \tan^{-1}(x) + C$ (xi) $\int e^x \, dx = e^x + C$ (xii) $\int \frac{1}{x} \, dx = \ln|x| + C$ (xiii) $\int a^x \, dx = \frac{a^x}{\ln(a)} + C$

Properties of Indefinite Integrals

The following properties govern the behavior of indefinite integrals:

Property I: The processes of differentiation and integration are inverses. Specifically, $\frac{d}{dx} \int f(x) \, dx = f(x)$ and $\int f'(x) \, dx = f(x) + C$. This is proven by letting $F$ be an anti-derivative such that $F'(x) = f(x)$, making $\int f(x) \, dx = F(x) + C$, and thus the derivative of the integral returns the integrand.

Property II: Two indefinite integrals with the same derivative lead to the same family of curves and are thus equivalent. If $\frac{d}{dx} \int f(x) \, dx = \frac{d}{dx} \int g(x) \, dx$, then $\int f(x) \, dx - \int g(x) \, dx = C$. This indicates that the families of curves represented by the two integrals are identical, which is written as $\int f(x) \, dx = \int g(x) \, dx$ without explicitly mentioning the differing parameters.

Property III: The integral of the sum of two functions is the sum of their integrals: $\int [f(x) + g(x)] \, dx = \int f(x) \, dx + \int g(x) \, dx$.

Property IV: For any real number $k$, $\int k f(x) \, dx = k \int f(x) \, dx$.

Property V: Properties III and IV can be generalized to a finite number of functions $f_1, f_2, \dots, f_n$ and real numbers $k_1, k_2, \dots, k_n$: $\int [k_1 f_1(x) + k_2 f_2(x) + \dots + k_n f_n(x)] \, dx = k_1 \int f_1(x) \, dx + k_2 \int f_2(x) \, dx + \dots + k_n \int f_n(x) \, dx$.

Integration Methods: Inspection, Substitution, and Identities

The method of inspection involves searching intuitively for a function whose derivative matches the integrand. For instance, to find an anti-derivative of $\cos(2x)$, we recall $\frac{d}{dx}(\sin(2x)) = 2\cos(2x)$, implying the anti-derivative is $\frac{1}{2}\sin(2x)$. To find the anti-derivative of $3x^2 + 4x^3$, we note $\frac{d}{dx}(x^3 + x^4) = 3x^2 + 4x^3$. For $\frac{1}{x}$ where $x \neq 0$, the anti-derivative is $\ln|x|$.

Integration by substitution involves transforming the integral $\int f(x) \, dx$ by changing the independent variable $x$ to $t$ via the substitution $x = g(t)$. This results in $dx = g'(t) \, dt$, transforming the integral to $\int f(g(t))g'(t) \, dt$. Usually, we substitute for a function whose derivative also appears in the integrand. Examples include:

1. For $\int \sin(mx) \, dx$, substituting $mx = t$ leads to $-\frac{1}{m}\cos(mx) + C$.
2. For $\int 2x \sin(x^2 + 1) \, dx$, substituting $x^2 + 1 = t$ leads to $-\cos(x^2 + 1) + C$.
3. The standard trigonometric integrals using substitution are:
   • $\int \tan(x) \, dx = \ln|\sec(x)| + C$
   • $\int \cot(x) \, dx = \ln|\sin(x)| + C$
   • $\int \sec(x) \, dx = \ln|\sec(x) + \tan(x)| + C$
   • $\int \text{cosec}(x) \, dx = \ln|\text{cosec}(x) - \cot(x)| + C$

Integration using trigonometric identities is necessary when the integrand involves powers or products of trig functions. Identities such as $\cos^2(x) = \frac{1 + \cos(2x)}{2}$, $\sin(x)\cos(y) = \frac{1}{2}[\sin(x + y) + \sin(x - y)]$, and $\sin(3x) = 3\sin(x) - 4\sin^3(x)$ are frequently employed to reduce the integrand to a standard form.

Integrals of Particular Functions

There are several standard formulae for integrating specific forms of rational and irrational functions:

1. $\int \frac{dx}{x^2 - a^2} = \frac{1}{2a} \ln\left|\frac{x - a}{x + a}\right| + C$
2. $\int \frac{dx}{a^2 - x^2} = \frac{1}{2a} \ln\left|\frac{a + x}{a - x}\right| + C$
3. $\int \frac{dx}{x^2 + a^2} = \frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right) + C$
4. $\int \frac{dx}{\sqrt{x^2 - a^2}} = \ln|x + \sqrt{x^2 - a^2}| + C$
5. $\int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}\left(\frac{x}{a}\right) + C$
6. $\int \frac{dx}{\sqrt{x^2 + a^2}} = \ln|x + \sqrt{x^2 + a^2}| + C$

To evaluate integrals of the type $\int \frac{dx}{ax^2 + bx + c}$, one must complete the square in the denominator to reduce it to one of the standard forms above. For integrals of type $\int \frac{px + q}{ax^2 + bx + c} \, dx$, the numerator is expressed as $A \frac{d}{dx}(ax^2 + bx + c) + B$. Equating coefficients allows one to find the constants $A$ and $B$, splitting the integral into a log form and a standard rational form.

Integration by Partial Fractions

If a rational function $P(x)/Q(x)$ is proper (degree of $P(x)$ < degree of $Q(x)$), it can be decomposed into simpler partial fractions based on the factors of the denominator $Q(x)$. If the function is improper, one must first use long division to express it as a polynomial plus a proper rational function.

Table 7.2 defines the decomposition forms:

1. $\frac{px + q}{(x - a)(x - b)} = \frac{A}{x - a} + \frac{B}{x - b}$
2. $\frac{px + q}{(x - a)^2} = \frac{A}{x - a} + \frac{B}{(x - a)^2}$
3. $\frac{px^2 + qx + r}{(x - a)(x - b)(x - c)} = \frac{A}{x - a} + \frac{B}{x - b} + \frac{C}{x - c}$
4. $\frac{px^2 + qx + r}{(x - a)^2(x - b)} = \frac{A}{x - a} + \frac{B}{(x - a)^2} + \frac{C}{x - b}$
5. $\frac{px^2 + qx + r}{(x - a)(x^2 + bx + c)} = \frac{A}{x - a} + \frac{Bx + C}{x^2 + bx + c}$, where the quadratic cannot be factorized.

Integration by Parts

This method is used for integrating the product of two functions $u$ and $v$. Derived from the product rule of differentiation, the formula is: $\int f(x)g(x) \, dx = f(x) \int g(x) \, dx - \int \left[ f'(x) \int g(x) \, dx \right] \, dx$ In words, the integral of the product equals the first function times the integral of the second, minus the integral of the derivative of the first times the integral of the second. The choice of the first function is significant; generally, $\ln(x)$ or inverse trigonometric functions are chosen as the first function, while polynomial or power functions are chosen as the first function if the other function is exponential or trigonometric. For $\int \ln(x) \, dx$, we treat it as $\int \ln(x) \cdot 1 \, dx$, yielding $x\ln(x) - x + C$.

A special type of integral is $\int e^x [f(x) + f'(x)] \, dx = e^x f(x) + C$. This is solved by splitting the integral and applying integration by parts to the first term.

Additional Special Integrals and Definite Integrals

Three more standard types based on integration by parts involve the square root of quadratic expressions:

1. $\int \sqrt{x^2 - a^2} \, dx = \frac{x}{2}\sqrt{x^2 - a^2} - \frac{a^2}{2} \ln|x + \sqrt{x^2 - a^2}| + C$
2. $\int \sqrt{x^2 + a^2} \, dx = \frac{x}{2}\sqrt{x^2 + a^2} + \frac{a^2}{2} \ln|x + \sqrt{x^2 + a^2}| + C$
3. $\int \sqrt{a^2 - x^2} \, dx = \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1}\left(\frac{x}{a}\right) + C$

Definite integrals are denoted by $\int_a^b f(x) \, dx$, where $a$ is the lower limit and $b$ is the upper limit. Unlike indefinite integrals, a definite integral has a unique value. It can be interpreted as the area function $A(x) = \int_a^x f(x) \, dx$, representing the area under the curve from $a$ to $x$.

Fundamental Theorem of Calculus Theorem 1 states that if $f$ is continuous on $[a, b]$, then $A'(x) = f(x)$. Theorem 2 (the evaluation theorem) states that $\int_a^b f(x) \, dx = F(b) - F(a)$, where $F$ is any anti-derivative of $f$. When using substitution for definite integrals, it is often more efficient to change the limits of integration according to the substitution variable rather than resubstituting the original variable at the end.

Properties of Definite Integrals

The following properties facilitate the calculation of definite integrals:

$P_0: \int_a^b f(x) \, dx = \int_a^b f(t) \, dt$$P_1: \int_a^b f(x) \, dx = -\int_b^a f(x) \, dx$. Specifically, $\int_a^a f(x) \, dx = 0$. $P_2: \int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx$$P_3: \int_a^b f(x) \, dx = \int_a^b f(a + b - x) \, dx$$P_4: \int_0^a f(x) \, dx = \int_0^a f(a - x) \, dx$$P_5: \int_0^{2a} f(x) \, dx = \int_0^a f(x) \, dx + \int_0^a f(2a - x) \, dx$$P_6: \int_0^{2a} f(x) \, dx = \begin{cases} 2\int_0^a f(x) \, dx & \text{if } f(2a - x) = f(x) \\ 0 & \text{if } f(2a - x) = -f(x) \end{cases}$$P_7: \int_{-a}^a f(x) \, dx = \begin{cases} 2\int_0^a f(x) \, dx & \text{if } f \text{ is even } (f(-x) = f(x)) \\ 0 & \text{if } f \text{ is odd } (f(-x) = -f(x)) \end{cases}$

Example cases include $\int_{-1}^1 \sin^5(x)\cos^4(x) \, dx = 0$ because the integrand is an odd function, and $\int_0^{\pi} \frac{x \sin(x)}{1 + \cos^2(x)} \, dx = \frac{\pi^2}{4}$, which is solved by using property $P_4$ to eliminate the $x$ term in the numerator.