Indefinite Integrals

The second part of the Fundamental Theorem of Calculus indicates a strong relationship between definite integrals and antiderivatives; we abuse this relationship to introduce a convenient notation for a function’s antiderivative. The indefinite integral of a function \(f,\) denoted \(\int f(x)\,\mathrm{d}x\) (without bounds), denotes the family of all antiderivatives of \(f.\) The definite integral of a function on the interval \([a,b]\) can be computed as the indefinite integral evaluated at the bounds \(x=a\) and \(x=b\) Symbolically, for \(f(x) = F'(x),\) \[ \int f(x)\,\mathrm{d}x \;=\; F(x) + C \qquad\qquad \int_a^b f(x)\,\mathrm{d}x \;=\; \biggl(\int f(x)\,\mathrm{d}x\biggr)\biggr\rvert_{x=a}^{x=b} %= \Bigl(F(x) + C\Bigr)\biggr\rvert_{x=a}^{x=b} \;=\; \Bigl(F(b)+C\Bigr)-\Bigl(F(a)+C\Bigr) \;=\; F(b)-F(a) \,. \]

This indefinite integral notation affords us a more “forward” way of writing antidifferentiation formulas:

\(\displaystyle \int x^{\color{Maroon}n} \,\mathrm{d}x = {\color{Maroon}\frac{1}{n+1}}x^{\color{Maroon}n+1} + C \)

The power rule for \(n \neq -1\)

\(\displaystyle \int x^{-1} \,\mathrm{d}x = \ln(x) + C \)

The power rule for \(n = -1\)

\[\begin{align*} \int {\color{DarkSlateBlue}\cos}(x) \,\mathrm{d}x &= {\color{DarkGoldenRod}\sin}(x) + C \\[1em] \int {\color{DarkGoldenRod}\sin}(x) \,\mathrm{d}x &= -{\color{DarkSlateBlue}\cos}(x) + C \end{align*}\]

\[\begin{align*} \int {\color{ForestGreen}\sec}(x){\color{Firebrick}\tan}(x) \,\mathrm{d}x &= {\color{ForestGreen}\sec}(x) + C \\[1em] \int {\color{DarkOliveGreen}\csc}(x){\color{Maroon}\cot}(x) \,\mathrm{d}x &= -{\color{DarkOliveGreen}\csc}(x) + C \end{align*}\]

\[\begin{align*} \int {\color{ForestGreen}\sec}^2(x) \,\mathrm{d}x &= {\color{Firebrick}\tan}(x) + C \\[1em] \int {\color{DarkOliveGreen}\csc}^2(x) \,\mathrm{d}x &= -{\color{Maroon}\cot}(x) + C \end{align*}\]