Skip to content

Mathematics and Statistics · Ch 6 — Definite Integration

Standard Definite Integrals and Linearity

2

Standard Definite Integrals and Linearity

Evaluating a definite integral needs the same list of standard antiderivatives used for indefinite integration, now applied between limits. The results below cover every integrand type met in this course.

Power rule (for n≠−1n \neq -1):

∫abxn dx=[xn+1n+1]ab=bn+1−an+1n+1.\int_{a}^{b} x^{n}\,dx = \left[\frac{x^{n+1}}{n+1}\right]_{a}^{b} = \frac{b^{n+1} - a^{n+1}}{n+1}.

Reciprocal (the n=−1n = -1 case, for 0<a<b0 < a < b):

∫ab1x dx=[log⁡∣x∣]ab=log⁡b−log⁡a=log⁡ ⁣ba.\int_{a}^{b} \frac{1}{x}\,dx = \big[\log|x|\big]_{a}^{b} = \log b - \log a = \log\!\frac{b}{a}.

Exponential:

∫abex dx=[ex]ab=eb−ea.\int_{a}^{b} e^{x}\,dx = \big[e^{x}\big]_{a}^{b} = e^{b} - e^{a}.

Constant integrand (for a constant kk):

∫abk dx=[kx]ab=k(b−a).\int_{a}^{b} k\,dx = \big[kx\big]_{a}^{b} = k(b - a).

Linearity of the definite integral. For constants α,β\alpha, \beta and integrable functions f,gf, g:

∫ab[αf(x)+βg(x)] dx=α∫abf(x) dx+β∫abg(x) dx.\int_{a}^{b}\big[\alpha f(x) + \beta g(x)\big]\,dx = \alpha\int_{a}^{b} f(x)\,dx + \beta\int_{a}^{b} g(x)\,dx.

This lets a sum or difference be integrated term by term, with constant multipliers pulled outside the integral sign. Almost every polynomial integral in this chapter is evaluated by combining the power rule with linearity.

Illustration. ∫12(x2+3) dx=∫12x2 dx+∫123 dx=[x33]12+[3x]12=8−13+3(2−1)=73+3=163.\displaystyle\int_{1}^{2}\big(x^{2} + 3\big)\,dx = \int_{1}^{2} x^{2}\,dx + \int_{1}^{2} 3\,dx = \left[\frac{x^{3}}{3}\right]_{1}^{2} + \big[3x\big]_{1}^{2} = \frac{8-1}{3} + 3(2-1) = \frac{7}{3} + 3 = \frac{16}{3}.

Note

Substitute the Upper Limit First, Then Subtract the Lower — Mind the Signs …

Definition 3Standard definite integrals

∫abxn dx=bn+1−an+1n+1\displaystyle\int_a^b x^n\,dx = \frac{b^{n+1}-a^{n+1}}{n+1} (n≠−1n\neq-1); ∫ab1x dx=log⁡ba\displaystyle\int_a^b \tfrac1x\,dx = \log\tfrac{b}{a} (0<a<b0<a<b); ∫abex dx=eb−ea\displaystyle\int_a^b e^x\,dx = e^b-e^a; $\dis …

Definition 4Linearity

∫ab(αf+βg) dx=α∫abf dx+β∫abg dx\displaystyle\int_a^b(\alpha f + \beta g)\,dx = \alpha\int_a^b f\,dx + \beta\int_a^b g\,dx: a definite integral of a sum is integrated term by term, and constant multipliers …

Start your 14-day free trial to unlock the full solution →
← Previous · 1The Definite Integral and the Fundamental TheoremNext · 3 →Properties of Definite Integrals