Business Mathematics and Statistics · Ch 7 — Integration

Standard Integrals

This chapter fixes a small set of standard integrals as the toolkit every later worked example draws from — each one provable, and checkable, by differentiating the right-hand side.

Linearity of integration

Before listing the formulas, one property makes them usable together: integration is linear — the integral of a sum (or difference) is the sum (or difference) 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 term by term, one standard formula at a time.

The standard integrals used in this chapter

#	Integral	Result	Condition
1	$\displaystyle\int x^{n}\,dx$	$\dfrac{x^{n+1}}{n+1}+C$	$n \ne -1$
2	$\displaystyle\int \frac{1}{x}\,dx$	$\ln	x
3	$\displaystyle\int e^{x}\,dx$	$e^{x}+C$	—
4	$\displaystyle\int k\,dx$ (a constant $k$ )	$kx+C$	—

Note

Why formula 1 excludes $n=-1$

Substituting $n=-1$ into $\dfrac{x^{n+1}}{n+1}$ gives $\dfrac{x^{0}}{0}$ , undefined by division by zero — this is exactly the case formula 2 exists to cover separately, since $\displaystyle\int x^{-1}\,dx = \int \frac{1}{x}\,dx = \ln|x|+C$ .

Worked pattern combining several terms …

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 2The Standard Integrals

$\int x^{n}dx=\frac{x^{n+1}}{n+1}+C\ (n\ne-1)$ ; $\int\frac{1}{x}dx=\ln|x|+C$ ; $\int e^{x}dx=e^{x}+C$ ; $\int k,dx=kx+C …

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