### Differentials and integrals: Indefinite integrals

### Antiderivatives of standard functions

The search for antiderivative of a given function on an interval, that is, the search for a function \(F(x)\) for a given function \(f(x)\) on an interval \(I\) with the property that \(F'(x)=f(x)\), is called **finding the primitive** or **integrating.** The following equality is valid for this primitive function \(F\) and any \(a\) and \(x\) in \(I\) \[F(x)=F(a)+ \int_a^x f(\xi)\,\dd \xi.\] Every **antiderivative** (also called **primitive**) of \(f(x)\) on \(I\) can be written this way. One often denotes such a primitive function of \(f(x)\) without lower and upper limits in the format \[\int f(x)\,\dd x\] and then one callls it an **indefinite integral.** The indeterminacy lies in the fact that antiderivative of \(f(x)\) is only uniquely determined up to a **constant of integration**. A simple example of an indefinite integral on \(I=(-\infty,\infty)\) is \[\int (2x+3)\,\dd x = x^2+3x+c\] where the constant of integration is denoted with the letter \(c\). Many mathematical software programs such as the computer algebra systems Maple and *Mathematica* do not mention the constant of integration and provide only one primitive function.

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Mathcentre video

Integration as the reverse of differentiation (34:14)