What is the antiderivative (indefinite integral) and how to find it?

Understanding Antiderivatives (Indefinite Integrals)

An antiderivative of a function, also known as an indefinite integral, is a function whose derivative is the original function. In other words, if you differentiate an antiderivative, you get the original function back. The antiderivative is a way to reverse the operation of differentiation.

Notation and Basic Rules

• The antiderivative of a function $f(x)$ is denoted as $\int f(x) \, dx$.
• The $dx$ is a part of the notation and is not something you do to $f(x)$. It's called the differential of $x$.
• The antiderivative is not unique; it's defined up to a constant. This constant is often denoted as $C$.

How to Find Antiderivatives

1. Power Rule: If $f(x) = x^n$, then $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$, where $n \neq -1$.
2. Constant Rule: If $f(x) = k$, where $k$ is a constant, then $\int k \, dx = kx + C$.
3. Sum and Difference Rules: The antiderivative of a sum or difference of functions is the sum or difference of their antiderivatives.
   • $\int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx$
   • $\int (f(x) - g(x)) \, dx = \int f(x) \, dx - \int g(x) \, dx$
4. Substitution (u-substitution): This is a more advanced technique used to simplify integrals by making a substitution that makes the integral easier to evaluate. The process involves finding a $u$ such that $du = f(x) \, dx$, then solving for $x$ in terms of $u$, and finally substituting this expression into the antiderivative.