Indefinite integrals of common functions

Indefinite integrals, also known as antiderivatives, represent a family of functions whose derivative gives the original function. The concept revolves around finding the integral of common functions, which is expressed in the form:

\int f(x) \, dx = F(x) + C

where $F(x)$ is the antiderivative of $f(x)$ , and $C$ is the constant of integration.

Here are some basic indefinite integrals of common functions:

Power Function:
$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$
Exponential Function:
$\int e^x \, dx = e^x + C$
Trigonometric Functions:
- Sine: $\int \sin(x) \, dx = -\cos(x) + C$
- Cosine: $\int \cos(x) \, dx = \sin(x) + C$
- Tangent: $\int \tan(x) \, dx = -\ln|\cos(x)| + C$
Natural Logarithm:
$\int \frac{1}{x} \, dx = \ln|x| + C \quad (x \neq 0)$
Special Cases:
- Integrating constants: $\int a \, dx = ax + C$
- For $f(x) = 0$ : $\int 0 \, dx = C$

Indefinite integrals play a crucial role in calculus, providing the foundations for solving differential equations and understanding the area under curves.

In this article

Part 1: Indefinite integral of 1/x Part 2: Indefinite integrals of sin(x), cos(x), and eˣ

Part 1: Indefinite integral of 1/x

In differential calculus we learned that the derivative of ln(x) is 1/x. Integration goes the other way: the integral (or antiderivative) of 1/x should be a function whose derivative is 1/x. As we just saw, this is ln(x). However, if x is negative then ln(x) is undefined! The solution is quite simple: the antiderivative of 1/x is ln(|x|).

Here are the key points for studying the indefinite integral of $\frac{1}{x}$ :

Basic Formula: The indefinite integral of $\frac{1}{x}$ is given by:
$\int \frac{1}{x} \, dx = \ln |x| + C$
where $C$ is the constant of integration.
Natural Logarithm: The result involves the natural logarithm function $\ln$ , emphasizing the importance of understanding logarithmic properties.
Absolute Value: The absolute value $|x|$ is used because the logarithm function is only defined for positive arguments, ensuring the integral is valid for both positive and negative values of $x$ (except $x = 0$ ).
Domain Considerations: Recognize that the integral is not defined at $x = 0$ , which creates a discontinuity.
Applications: Understanding this integral is crucial for solving various problems in calculus, including those involving logarithmic growth and decay, as well as integration techniques.
Integration by Substitution: Recognize that integrals involving transformations or substitutions can lead back to this fundamental result.

These points are essential for a solid understanding of integrating $\frac{1}{x}$ .

Part 2: Indefinite integrals of sin(x), cos(x), and eˣ

∫sin(x)dx=-cos(x)+C, ∫cos(x)dx=sin(x)+C, and ∫eˣdx=eˣ+C. Learn why this is so and see worked examples.

When studying the indefinite integrals of $\sin(x)$ , $\cos(x)$ , and $e^x$ , focus on the following key points:

Basic Integration Formulas:
- The integral of $\sin(x)$ is: $\int \sin(x) \, dx = -\cos(x) + C$
- The integral of $\cos(x)$ is: $\int \cos(x) \, dx = \sin(x) + C$
- The integral of $e^x$ is: $\int e^x \, dx = e^x + C$
Constant of Integration:
- Always include the constant of integration $C$ in your results, as indefinite integrals represent a family of functions.
Fundamental Theorem of Calculus:
- Understand how indefinite integrals relate to the definite integrals. The process of differentiation and integration are inverse operations.
Trigonometric Relationships:
- Be familiar with trigonometric identities, as they can be useful in more complex problems involving these functions.
Applications:
- Recognize the practical application of these integrals in solving problems related to areas under curves and in physics.

By mastering these key points, you'll have a solid understanding of the indefinite integrals of $\sin(x)$ , $\cos(x)$ , and $e^x$ .

Indefinite integrals of common functions

Part 1: Indefinite integral of 1/x

Part 2: Indefinite integrals of sin(x), cos(x), and eˣ

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