Indefinite integrals

Definition

An indefinite integral, also known as an antiderivative, is a function that reverses the process of differentiation. It represents a family of functions whose derivative is the given function.

5 Must Know Facts For Your Next Test

1. The general form of an indefinite integral is $\int f(x) \, dx = F(x) + C$, where $F(x)$ is the antiderivative of $f(x)$ and $C$ is the constant of integration.
2. Integration by parts is a technique used to find indefinite integrals and follows the formula: $$\int u \, dv = uv - \int v \, du$$.
3. Choosing $u$ and $dv$ correctly in integration by parts can simplify the problem significantly; typically, $u$ is chosen to be a function that simplifies upon differentiation.
4. $e^x$, trigonometric functions like $\sin x$ and $\cos x$, and polynomials are common functions encountered in indefinite integrals.
5. Indefinite integrals are essential for solving differential equations, which model many real-world phenomena.

Review Questions

• What is the general form of an indefinite integral?
• Explain how you would choose $u$ and $dv$ in the method of integration by parts.
• What role does the constant of integration play in an indefinite integral?

Related terms

Definite Integral: A definite integral calculates the net area under a curve over a specified interval, represented as $\int_a^b f(x) \, dx$.

Integration by Parts: A technique for integrating products of functions using the formula: $$\int u \, dv = uv - \int v \, du$$.

Antiderivative: Another term for an indefinite integral; it represents a function whose derivative is the given function.