Properties of Definite Integrals

These eight properties let a definite integral be simplified — often to zero, or to twice a simpler integral, or to another definite integral that is easier to handle — purely by symmetry or by a change of variable, without ever writing down the antiderivative explicitly. In every proof below, $g$ denotes any primitive of $f$, i.e. $\int f(x)\,dx=g(x)+c$, and the Fundamental Theorem (Section 4.2) is used to convert each bracketed evaluation.

Property I. $\displaystyle\int_a^a f(x)\,dx = 0$.

Proof: $\int_a^a f(x)\,dx=[g(x)+c]_a^a=(g(a)+c)-(g(a)+c)=0$ — the upper and lower limit are the same point, so the bracket has nothing to measure.

Property II. $\displaystyle\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx$.

Proof: $\int_a^b f(x)\,dx=g(b)-g(a)=-\big(g(a)-g(b)\big)=-\int_b^a f(x)\,dx$. Swapping the limits of a definite integral flips its sign.

Worked examples: $\int_3^1 x\,dx=\left[\frac{x^2}2\right]_3^1=\frac12-\frac92=-4$, while $\int_1^3 x\,dx=\left[\frac{x^2}2\right]_1^3=\frac92-\frac12=4$ — exactly the negative of each other, confirming the property.

Property III. $\displaystyle\int_a^b f(x)\,dx = \int_a^b f(t)\,dt$.

Proof: both sides reduce, via the Fundamental Theorem, to $g(b)-g(a)$ regardless of whether the dummy integration variable is called $x$ or $t$. This says definite integration is independent of the name given to the variable — the definite integral is a fixed number determined only by $f$, $a$, and $b$, not by the symbol used inside. Worked example: both $\int_{\pi/6}^{\pi/3}\cos x\,dx$ and $\int_{\pi/6}^{\pi/3}\cos t\,dt$ evaluate to $[\sin(\cdot)]_{\pi/6}^{\pi/3}=\sin\frac\pi3-\sin\frac\pi6=\frac{\sqrt3}2-\frac12=\frac{\sqrt3-1}2$.

Property IV (additivity over sub-intervals). $\displaystyle\int_a^b f(x)\,dx = \int_a^c f(x)\,dx+\int_c^b f(x)\,dx$, for any $c\in[a,b]$.

Proof: the right side is $\big(g(c)-g(a)\big)+\big(g(b)-g(c)\big)=g(b)-g(a)$, which is exactly the left side. Worked example: for $\int_{-1}^5(2x+3)\,dx$ with $c=3$, both the direct evaluation $[x^2+3x]_{-1}^5=(25+15)-(1-3)=42$ and the split evaluation $[x^2+3x]_{-1}^3+[x^2+3x]_3^5=\big[(9+9)-(1-3)\big]+\big[(25+15)-(9+9)\big]=20+22=42$ agree.

Property V. $\displaystyle\int_a^b f(x)\,dx = \int_a^b f(a+b-x)\,dx$.

Proof: substitute $t=a+b-x$ in the right side, so $x=a+b-t$, $dx=-dt$; as $x$ runs from $a$ to $b$, $t$ runs from $b$ down to $a$. So $\int_a^b f(a+b-x)\,dx=\int_b^a f(t)(-dt)=-\int_b^a f(t)\,dt=\int_a^b f(t)\,dt$ (using Property II), which by Property III equals $\int_a^b f(x)\,dx$. This is the single most useful property in the chapter: reflecting $x$ about the midpoint of $[a,b]$ never changes the value of the integral.

Worked example: for $I=\int_{\pi/6}^{\pi/3}\sin^2x\,dx$, reflecting via $x\to\frac\pi6+\frac\pi3-x=\frac\pi2-x$ turns $\sin^2x$ into $\sin^2\!\big(\frac\pi2-x\big)=\cos^2x$, so $I=\int_{\pi/6}^{\pi/3}\cos^2x\,dx$ too. Adding the two expressions for $I$: $2I=\int_{\pi/6}^{\pi/3}(\sin^2x+\cos^2x)\,dx=\int_{\pi/6}^{\pi/3}1\,dx=\frac\pi3-\frac\pi6=\frac\pi6$, so $I=\frac\pi{12}$.

Property VI. $\displaystyle\int_0^a f(x)\,dx = \int_0^a f(a-x)\,dx$ — the special case of Property V when the lower limit is $0$ (so $a+b-x$ becomes $a-x$).

Worked example: for $I=\int_0^{\pi/4}\log(1+\tan x)\,dx$, reflecting via $x\to\frac\pi4-x$ and using $\tan\!\big(\frac\pi4-x\big)=\dfrac{1-\tan x}{1+\tan x}$ gives $1+\tan\!\big(\frac\pi4-x\big)=\dfrac{2}{1+\tan x}$, so $I=\int_0^{\pi/4}\big[\log2-\log(1+\tan x)\big]dx=\big(\log2\big)\cdot\frac\pi4-I$. Solving, $2I=\frac\pi4\log2$, so $I=\frac\pi8\log2$.

Property VII. $\displaystyle\int_0^{2a} f(x)\,dx = \int_0^a f(x)\,dx+\int_0^a f(2a-x)\,dx$.

Proof: in the second term on the right, substitute $t=2a-x$ ($dx=-dt$; $x:0\to a$ gives $t:2a\to a$), so $\int_0^a f(2a-x)\,dx=\int_{2a}^a f(t)(-dt)=\int_a^{2a}f(t)\,dt$. Adding $\int_0^a f(x)\,dx+\int_a^{2a}f(x)\,dx$ gives exactly $\int_0^{2a}f(x)\,dx$ by Property IV. This is the tool used whenever the upper limit is twice the useful reflection point.

Property VIII (odd/even functions).

$\int_{-a}^a f(x)\,dx = \begin{cases} 2\displaystyle\int_0^a f(x)\,dx, & f \text{ even } (f(-x)=f(x))\\[4pt] 0, & f \text{ odd } (f(-x)=-f(x)) \end{cases}$ …