Integration - Indefinite Integrals by Inspection

Properties of indefinite integrals

Theorem (Linearity property of indefinite integrals)

Let \(\mathrm{f}\) and \(\mathrm{g}\) be functions, and \(a, b\) constants. Then the following hold:
\[ \displaystyle \displaystyle\int \big(a \mathrm{f}(x) + b \mathrm{g}(x) \big) \, \mathrm{d}x = a \displaystyle \displaystyle\int \mathrm{f}(x) \, \mathrm{d}x + b \displaystyle \displaystyle\int \mathrm{g}(x) \, \mathrm{d}x \]

Proof.

Differentiate the expression
\[ a \displaystyle \displaystyle\int \mathrm{f}(x) \, {\rm d}x + b \displaystyle \displaystyle\int \mathrm{g}(x) \, {\rm d}x \]
with respect to \(x\) yields:
\[ \begin{align*} \frac{\mathrm{d}}{\mathrm{d}x} \big( a \displaystyle \displaystyle\int \mathrm{f}(x) \, {\mathrm d}x + b \displaystyle \displaystyle\int \mathrm{g}(x) \, \mathrm{d}x \big) & = a \big( \frac{\mathrm{d}}{\mathrm{d}x} \displaystyle \displaystyle\int \mathrm{f}(x) \, {\mathrm d}x \big) + b \big( \frac{\mathrm{d}}{\mathrm{d}x} \displaystyle \displaystyle\int \mathrm{g}(x) \, {\mathrm d}x \big) \\ & = a \mathrm{f}(x) + b \mathrm{g}(x). \end{align*} \]
Hence, \(\displaystyle a \displaystyle \displaystyle\int \mathrm{f}(x) \, {\mathrm d}x + b \displaystyle \displaystyle\int \mathrm{g}(x) \, {\mathrm d}x\) is an antiderivative of \(a \mathrm{f}(x) + b \mathrm{g}(x)\).

\(\Box\)

Corollary (Scalar multiple)

Let \(\mathrm{f}\) be a function and \(k\) a constant. Then
\[ \displaystyle \displaystyle\int k \mathrm{f}(x) \, \mathrm{d}x = k \displaystyle \displaystyle\int \mathrm{f}(x) \, \mathrm{d}x. \]

Proof. Applying the above theorem by setting \(a = k\) and \(b=0\).

\(\Box\)