Mathematics and Statistics · Ch 5 — Integration

Standard Integrals and Rules of Integration

This section fixes the basic toolkit of standard integrals and the two rules that let them be combined. Each standard result is simply a known derivative read backwards, so each can be confirmed by differentiating its right-hand side.

Rules of integration (linearity)

Integration is linear — the integral of a sum is the sum of the integrals, and a constant multiplier can be pulled outside the integral sign:

\int \big[a\,f(x) + b\,g(x)\big]\,dx = a\int f(x)\,dx + b\int g(x)\,dx.

This is exactly what lets a multi-term expression be integrated one term at a time.

The standard integrals

#	Integral	Result	Condition
1	$\displaystyle\int x^{n}\,dx$	$\dfrac{x^{n+1}}{n+1}+c$	$n \ne -1$
2	$\displaystyle\int \frac{1}{x}\,dx$	$\log\lvert x\rvert+c$	$x \ne 0$
3	$\displaystyle\int e^{x}\,dx$	$e^{x}+c$	—
4	$\displaystyle\int a^{x}\,dx$	$\dfrac{a^{x}}{\log a}+c$	$a>0,\ a\ne 1$
5	$\displaystyle\int k\,dx$	$kx+c$	$k$ constant

Note

Why formula 1 excludes $n=-1$

Substituting $n=-1$ into $\dfrac{x^{n+1}}{n+1}$ gives $\dfrac{x^{0}}{0}$ , which is undefined (division by zero). That case is exactly what formula 2 covers separately, since $\displaystyle\int x^{-1}\,dx = \int \frac{1}{x}\,dx = \log\lvert x\rvert+c$ .

Verifying formula 4 by differentiation …

Definition 1Linearity of Integration

$\displaystyle\int\big[a\,f(x)+b\,g(x)\big]\,dx = a\!\int f(x)\,dx + b\!\int g(x)\,dx$ — a sum can be integrated term by term, and a constant factor can be mov …

Definition 2Basic Standard Integrals

$\int x^{n}dx=\frac{x^{n+1}}{n+1}+c\ (n\ne-1)$ ; $\int\frac{1}{x}dx=\log|x|+c$ ; $\int e^{x}dx=e^{x}+c$ ; $\int a^{x}dx=\frac{a^{x}}{\log a …

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