Indefinite integrals of common functions

Indefinite integrals, also known as antiderivatives, represent the family of functions whose derivative yields the original function. The indefinite integral of a function $f(x)$ is denoted as $\int f(x) \, dx$ and includes a constant of integration $C$ , since the derivative of a constant is zero.

Here are some common indefinite integrals:

Power Rule:
$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$
Exponential Functions:
$\int e^x \, dx = e^x + C$ $\int a^x \, dx = \frac{a^x}{\ln(a)} + C \quad (a > 0, a \neq 1)$
Trigonometric Functions:
$\int \sin(x) \, dx = -\cos(x) + C$ $\int \cos(x) \, dx = \sin(x) + C$ $\int \sec^2(x) \, dx = \tan(x) + C$
Natural Logarithm:
$\int \frac{1}{x} \, dx = \ln|x| + C \quad (x \neq 0)$

These rules are foundational for solving integrals in calculus, providing a systematic approach for finding antiderivatives of common functions. Each integral expresses a general family of functions that differ by a constant.

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|).

When studying the indefinite integral of $\frac{1}{x}$ , focus on these key points:

Integral Expression: The integral is expressed as:
$\int \frac{1}{x} \, dx$
Natural Logarithm: The result of the integral is:
$\int \frac{1}{x} \, dx = \ln |x| + C$
where $C$ is the constant of integration.
Domain Considerations: The function $\frac{1}{x}$ is undefined at $x = 0$ , so the integral is valid for $x \neq 0$ .
Absolute Value: The absolute value is important because the natural logarithm is only defined for positive arguments. This allows the integral to be valid for both positive and negative values of $x$ .
Applications: Indefinite integrals of $\frac{1}{x}$ often arise in various fields such as physics, economics, and engineering, particularly in problems involving growth rates and logarithmic scales.
Integration Techniques: Recognize that this integral is a standard form and can be derived from the rules of basic integration.

Remember these points for a solid understanding of the indefinite integral of $\frac{1}{x}$ .

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.

Here are the key points for studying the indefinite integrals of $\sin(x)$ , $\cos(x)$ , and $e^x$ :

Indefinite Integral of $\sin(x)$ :
- The integral is given by: $\int \sin(x) \, dx = -\cos(x) + C$
- The antiderivative is the negative cosine function plus a constant $C$ .
Indefinite Integral of $\cos(x)$ :
- The integral is given by: $\int \cos(x) \, dx = \sin(x) + C$
- The antiderivative is the sine function plus a constant $C$ .
Indefinite Integral of $e^x$ :
- The integral is given by: $\int e^x \, dx = e^x + C$
- The antiderivative of $e^x$ is itself, plus a constant $C$ .
Constant of Integration:
- Remember to include the constant $C$ in all indefinite integrals, representing the family of antiderivatives.
Understanding the Functions:
- Knowing the properties and graphs of $\sin(x)$ , $\cos(x)$ , and $e^x$ can help visualize their integrals and the behavior of their antiderivatives.

By focusing on these fundamental integrations, one can build a solid foundation for tackling more complex integrals involving these functions.

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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Proof videos