Glossary

7.2 Basic Integration Rules

3 min read•august 7, 2024

Integration rules are the building blocks of calculus, helping us reverse the process of differentiation. These rules allow us to find antiderivatives of functions, which is crucial for solving many real-world problems.

From constant and power rules to more complex techniques like , these tools give us the power to tackle a wide range of integrals. Mastering these rules is key to becoming proficient in calculus and unlocking its applications.

Basic Integration Rules

Constant and Power Rules

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  • states that the of a constant is the constant multiplied by the plus a (∫kdx=kx+C\int k dx = kx + C)
  • for integration states that the indefinite integral of a variable raised to a power is the variable raised to one more than the original power, divided by the new power, plus a constant of integration (∫xndx=xn+1n+1+C\int x^n dx = \frac{x^{n+1}}{n+1} + C)
    • For example, ∫x3dx=x44+C\int x^3 dx = \frac{x^4}{4} + C
    • The power rule applies to any real number power except -1 (∫1xdx=ln⁡∣x∣+C\int \frac{1}{x} dx = \ln|x| + C)

Constant Multiple, Sum, and Difference Rules

  • states that the indefinite integral of a constant multiplied by a function is equal to the constant multiplied by the indefinite integral of the function (∫kf(x)dx=k∫f(x)dx\int kf(x) dx = k \int f(x) dx)
    • For example, ∫3x2dx=3∫x2dx=3(x33)+C=x3+C\int 3x^2 dx = 3 \int x^2 dx = 3 (\frac{x^3}{3}) + C = x^3 + C
  • states that the indefinite integral of a sum of functions is equal to the sum of the indefinite integrals of each function (∫[f(x)+g(x)]dx=∫f(x)dx+∫g(x)dx\int [f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx)
    • For example, ∫(x2+3x)dx=∫x2dx+∫3xdx=x33+3x22+C\int (x^2 + 3x) dx = \int x^2 dx + \int 3x dx = \frac{x^3}{3} + \frac{3x^2}{2} + C
  • states that the indefinite integral of a difference of functions is equal to the difference of the indefinite integrals of each function (∫[f(x)−g(x)]dx=∫f(x)dx−∫g(x)dx\int [f(x) - g(x)] dx = \int f(x) dx - \int g(x) dx)

Transcendental Function Integration

Exponential and Logarithmic Function Rules

  • states that the indefinite integral of exe^x is itself plus a constant of integration (∫exdx=ex+C\int e^x dx = e^x + C)
    • More generally, the indefinite integral of an exponential function with base aa is 1ln⁡aax+C\frac{1}{\ln a} a^x + C
  • states that the indefinite integral of 1x\frac{1}{x} is the natural logarithm of the absolute value of xx plus a constant of integration (∫1xdx=ln⁡∣x∣+C\int \frac{1}{x} dx = \ln|x| + C)
    • This rule is a special case of the power rule where n=−1n = -1

Trigonometric Function Rules

  • The indefinite integrals of the basic trigonometric functions are as follows:
    • ∫sin⁡xdx=−cos⁡x+C\int \sin x dx = -\cos x + C
    • ∫cos⁡xdx=sin⁡x+C\int \cos x dx = \sin x + C
    • ∫sec⁡2xdx=tan⁡x+C\int \sec^2 x dx = \tan x + C
    • ∫csc⁡2xdx=−cot⁡x+C\int \csc^2 x dx = -\cot x + C
    • ∫sec⁡xtan⁡xdx=sec⁡x+C\int \sec x \tan x dx = \sec x + C
    • ∫csc⁡xcot⁡xdx=−csc⁡x+C\int \csc x \cot x dx = -\csc x + C

Advanced Integration Techniques

Integration by Substitution

  • Integration by substitution is a technique for finding antiderivatives of composite functions by substituting a new variable to simplify the integration process
  • The substitution method follows these steps:
    1. Choose a substitution variable uu as a function of xx that simplifies the
    2. Differentiate uu with respect to xx to find dudu
    3. Replace the original integrand with expressions involving uu and dudu
    4. Integrate the new expression with respect to uu
    5. Substitute back the original variable xx
  • For example, to find ∫xx2+1dx\int x \sqrt{x^2 + 1} dx, let u=x2+1u = x^2 + 1, then du=2xdxdu = 2x dx or xdx=12dux dx = \frac{1}{2} du, substituting yields ∫12u1/2du=13u3/2+C=13(x2+1)3/2+C\int \frac{1}{2} u^{1/2} du = \frac{1}{3} u^{3/2} + C = \frac{1}{3} (x^2 + 1)^{3/2} + C

Key Terms to Review (23)

Additivity of Integrals: The additivity of integrals states that the integral of a function over an interval can be expressed as the sum of integrals over subintervals that partition the original interval. This concept is crucial because it allows for breaking down complex integrals into simpler components, making it easier to compute definite integrals and understand their properties. By applying this principle, one can analyze functions more effectively and utilize it in conjunction with basic integration rules.
Constant multiple rule: The constant multiple rule states that when taking the derivative or integral of a function multiplied by a constant, you can factor the constant out of the operation. This rule is crucial for simplifying calculations in integration and differentiation, allowing for easier manipulation of functions. Understanding this rule helps in grasping how constants affect integrals and derivatives, making it foundational in calculus.
Constant of integration: The constant of integration is a term added to the result of an indefinite integral, representing an infinite set of possible functions that differ by a constant. It highlights that when integrating a function, we cannot determine the exact value of the function without additional information, as different constant values yield different functions. This concept is crucial in understanding the fundamental theorem of calculus and helps maintain the generality of integral solutions.
Constant Rule: The constant rule states that the derivative of a constant is zero, meaning if you have a function that is a constant value, its rate of change is zero regardless of the variable. This principle helps in simplifying the differentiation process and plays a significant role in integration as well. Understanding this rule is essential when working with limits, as constants maintain their value through transformations and operations.
Difference Rule: The difference rule is a fundamental principle in calculus that states the derivative of the difference of two functions is equal to the difference of their derivatives. This rule allows for the simplification of the differentiation process, making it easier to handle functions that are expressed as the subtraction of two or more terms. Understanding this rule is essential for mastering differentiation and is a key building block for more complex calculus concepts.
Exponential Function Rule: The exponential function rule is a fundamental principle in calculus that states the integral of an exponential function with a base of 'e' can be computed easily. Specifically, the rule states that for any real number 'a', the integral of the function $$a^x$$ with respect to 'x' is given by $$\frac{a^x}{\ln(a)} + C$$, where 'C' represents the constant of integration. This rule simplifies the process of integrating exponential functions, making it a key concept in understanding basic integration techniques.
Indefinite integral: An indefinite integral represents a family of functions whose derivative gives the original function. It is essentially the reverse process of differentiation and includes a constant of integration, usually denoted as 'C', because differentiating a constant results in zero. This concept is crucial as it connects to antiderivatives, basic integration rules, and methods like integration by parts, providing a foundational tool for solving various problems in calculus.
Integral of 1/x: The integral of 1/x is a fundamental concept in calculus, representing the area under the curve of the function f(x) = 1/x. This integral is unique because it leads to the natural logarithm function, specifically $$\int \frac{1}{x} \, dx = \ln |x| + C$$, where C is the constant of integration. Understanding this integral is essential as it lays the groundwork for more complex integration techniques and connects to various applications in mathematics and science.
Integral of cos(x): The integral of cos(x) represents the area under the curve of the cosine function over a given interval. This integral is fundamental in calculus, showcasing the relationship between differentiation and integration, particularly for trigonometric functions. Understanding how to compute the integral of cos(x) is crucial for solving problems that involve oscillatory behavior and periodic phenomena.
Integral of csc(x)cot(x): The integral of csc(x)cot(x) refers to the process of finding the antiderivative of the function csc(x)cot(x). This integral is particularly significant because it directly relates to the properties of trigonometric functions and their derivatives, showcasing how integration can reverse differentiation. Understanding this integral helps in mastering basic integration techniques and recognizing how certain functions can be simplified through their identities.
Integral of csc^2(x): The integral of csc^2(x) is a mathematical expression that represents the area under the curve of the cosecant squared function. This integral is particularly important because it directly relates to trigonometric functions, leading to the discovery of the tangent function, which is its antiderivative. Understanding this integral can help in solving various problems related to calculus and trigonometric identities.
Integral of e^x: The integral of e^x refers to the process of finding the antiderivative of the exponential function e^x, which is a fundamental concept in calculus. This integral is unique because the function e^x is its own derivative, meaning that when you integrate e^x, you simply get e^x plus a constant of integration. This property makes it an essential example in the basic rules of integration.
Integral of sec(x)tan(x): The integral of sec(x)tan(x) refers to the process of finding the antiderivative of the function sec(x)tan(x), which results in the function sec(x) plus a constant of integration. This integral is particularly significant in calculus as it demonstrates the relationship between trigonometric functions and their derivatives, highlighting how the integration process can often reverse differentiation.
Integral of sec^2(x): The integral of sec^2(x) is a fundamental concept in calculus, represented mathematically as $$\int \sec^2(x) \, dx = \tan(x) + C$$, where C is the constant of integration. This integral is particularly important because it arises frequently in problems involving trigonometric functions and is essential for solving differential equations related to these functions.
Integral of sin(x): The integral of sin(x) is a fundamental concept in calculus representing the area under the curve of the sine function. This integral is crucial in many applications across physics and engineering, especially in problems involving oscillations and waves. The result of this integral is related to the basic integration rules and trigonometric integrals, demonstrating how to compute areas and solve differential equations.
Integrand: An integrand is the function that is being integrated in the process of finding an integral. It is the core component of an integral expression and is typically denoted as f(x) when integrating with respect to x. The integrand plays a crucial role in determining the area under a curve, evaluating definite integrals, and simplifying complex integration processes.
Integration by substitution: Integration by substitution is a method used to simplify the process of finding integrals by transforming them into a more manageable form. This technique involves replacing a variable in the integral with another variable that represents a function of the original variable, often making the integral easier to evaluate. It connects deeply with the basic integration rules, allowing for more complex functions to be tackled efficiently.
Linearity of Integration: The linearity of integration refers to the property that allows for the integral of a sum of functions to be expressed as the sum of their individual integrals, and that a constant can be factored out of an integral. This principle is fundamental in calculus, as it simplifies the process of finding antiderivatives and evaluating definite integrals. Understanding this concept enables students to tackle more complex integrals and recognize patterns in integration techniques, especially when applying basic integration rules or working with trigonometric integrals.
Logarithmic Function Rule: The logarithmic function rule states that the derivative of a logarithmic function, specifically $$f(x) = ext{log}_b(u)$$ where $$u$$ is a differentiable function of $$x$$, is given by $$f'(x) = \frac{1}{u \ln(b)} \cdot u'$$. This rule highlights how the rate of change of a logarithmic function depends not only on its argument but also on the base of the logarithm, linking it to fundamental principles of calculus.
Power Rule: The power rule is a basic principle in calculus that provides a quick way to differentiate functions of the form $$f(x) = x^n$$, where $$n$$ is any real number. This rule states that the derivative of such a function is given by multiplying the coefficient by the exponent and then reducing the exponent by one, resulting in $$f'(x) = n \cdot x^{n-1}$$. This concept is foundational in understanding how to work with various types of functions, especially in the context of differentiation and integration.
Sum Rule: The sum rule is a fundamental principle in calculus that states the derivative or integral of a sum of functions is equal to the sum of their derivatives or integrals. This rule simplifies calculations, allowing for easier handling of complex expressions by breaking them down into simpler parts. Understanding this concept is crucial for mastering the basic rules of differentiation and integration, as it lays the groundwork for more advanced techniques in calculus.
Trigonometric Function Rules: Trigonometric function rules are a set of mathematical guidelines that define the relationships between the angles and lengths of triangles, primarily focusing on the six fundamental trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent. These rules are crucial for integrating functions that involve trigonometric expressions, as they allow for the manipulation and simplification of these functions during the integration process.
Variable of integration: The variable of integration is the symbol used in an integral to represent the input variable with respect to which the function is being integrated. It serves as a placeholder that indicates how the function changes concerning this specific variable. Understanding the role of the variable of integration is crucial as it helps clarify which function is being analyzed and ensures the correct application of integration rules.
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