Integration - Indefinite Integrals by Inspection

Indefinite Integrals

From differentiation, we know that
\[\frac{\mathrm{d}}{\mathrm{d}x}x^3=3x^2, \quad \frac{\mathrm{d}}{\mathrm{d}x}(x^3+7)=3x^2.\]

The reverse process of differentiation is called integration or anti-differentiation.

If the functions \(F(x)\) and \(\mathrm{f}(x)\) are related by \[\frac{\mathrm{d}}{\mathrm{d}x}F(x)=\mathrm{f}(x),\] we say that \(\mathrm{f}(x)\) is the derivative of \(\mathrm{F}(x)\), and \(\mathrm{F}(x)\) is an antiderivative of \(\mathrm{f}\).

Notation : An indefinite integral of \(\mathrm{f}(x)\) with respect to \(x\) is denoted by
\[\displaystyle \displaystyle\int \mathrm{f}(x) \, \mathrm{d}x = \mathrm{F}(x) + C,\]
where \(\mathrm{f}(x)\) is the integrand and \(C\) is an arbitrary constant of integration.

In the above example, an indefinite integral of \(f(x) = 3x^2\) can be either \((x^3+7)\) or \(x^3\), or more generally,
\[\mathrm{F}(x) = x^3 + C, \]
where \( C \) is any constant. Because all anti-derivatives of a given function differ by a constant, anti-derivatives are not unique.

Warning : It is easy to leave out the variable with respect to which the indefinite integral is to be performed, e.g., students often make mistake writing
\[ \displaystyle \displaystyle\int \mathrm{f}(x) \]
instead of the correct form
\[ \displaystyle \displaystyle\int \mathrm{f}(x) \, \mathrm{d}x. \]