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We use this fact and our knowledge of derivatives to find all the antiderivatives for several functions.
For each of the following functions, find all antiderivatives.
Find all antiderivatives of $f\left(x\right)=\text{sin}\phantom{\rule{0.1em}{0ex}}x.$
$\text{\u2212}\text{cos}\phantom{\rule{0.1em}{0ex}}x+C$
We now look at the formal notation used to represent antiderivatives and examine some of their properties. These properties allow us to find antiderivatives of more complicated functions. Given a function $f,$ we use the notation ${f}^{\prime}\left(x\right)$ or $\frac{df}{dx}$ to denote the derivative of $f.$ Here we introduce notation for antiderivatives. If $F$ is an antiderivative of $f,$ we say that $F\left(x\right)+C$ is the most general antiderivative of $f$ and write
The symbol $\int$ is called an integral sign , and $\int f\left(x\right)dx$ is called the indefinite integral of $f.$
Given a function $f,$ the indefinite integral of $f,$ denoted
is the most general antiderivative of $f.$ If $F$ is an antiderivative of $f,$ then
The expression $f\left(x\right)$ is called the integrand and the variable $x$ is the variable of integration .
Given the terminology introduced in this definition, the act of finding the antiderivatives of a function $f$ is usually referred to as integrating $f.$
For a function $f$ and an antiderivative $F,$ the functions $F\left(x\right)+C,$ where $C$ is any real number, is often referred to as the family of antiderivatives of $f.$ For example, since ${x}^{2}$ is an antiderivative of $2x$ and any antiderivative of $2x$ is of the form ${x}^{2}+C,$ we write
The collection of all functions of the form ${x}^{2}+C,$ where $C$ is any real number, is known as the family of antiderivatives of $2x.$ [link] shows a graph of this family of antiderivatives.
For some functions, evaluating indefinite integrals follows directly from properties of derivatives. For example, for $n\ne \text{\u2212}1,$
which comes directly from
This fact is known as the power rule for integrals .
For $n\ne \text{\u2212}1,$
Evaluating indefinite integrals for some other functions is also a straightforward calculation. The following table lists the indefinite integrals for several common functions. A more complete list appears in Appendix B .
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