3.1 Indefinite Integrals

If \(F(x)\) is an antiderivative of \(f(x)\), then \(F(x) + c\) is also an antiderivative of \(f(x)\) for every value of the constant \(c\) for it \(\dfrac {dF}{dx} = f\), then \begin {align*} \dfrac {d}{dx}\big [F + c\big ] & = \frac {d F}{dx} + \dfrac {d c}{dx}\\\\ & = f(x) + 0\\\\ & = f(x) \end {align*}

The set of all antiderivatives of a function \(f(x)\) is called the indefinite integral of \(f\) with respect to \(x\). The notation for the indefinite integral is \(\displaystyle {\int f(x)dx}\). When a formula \(F(x) + c\) gives all the antiderivatives of \(f\), we indicate this by writing \[\int f(x) dx = F(x) + c\]

The symbol \(\displaystyle {\int }\hspace {0.2cm}\) is an integral sign. The function \(f\) is an integrand and \(c\) is the constant of integration. The \(dx\) tells us that the variable of integration is \(x\).

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