The definition of a definite integral: ∫ a b f ( x ) d x {\displaystyle \int \limits _{a}^{b}f(x)dx} requires the interval [ a , b ] {\displaystyle [a,b]} be finite. The Fundamental Theorem of Calculus requires that f {\displaystyle f} be continuous on [ a , b ] {\displaystyle [a,b]} . In this section, you will be studying a method of evaluating integrals that fail these requirements—either because their limits of integration are infinite, or because a finite number of discontinuities exist on the interval [ a , b ] {\displaystyle [a,b]} . Integrals that fail either of these requirements are improper integrals. (If you are not familiar with L'Hôpital's rule, it is a good idea to review it before reading this section.) Consider the integral ∫ 1 ∞ d x x 2 {\displaystyle \int \limits _{1}^{\infty }{\frac {dx}{x^{2}}}} Assigning a finite upper bound b {\displaystyle b} in place of infinity gives lim b → ∞ ∫ 1 b d x x 2 = lim b → ∞ ( 1 1 − 1 b ) = lim b → ∞ ( 1 − 1 b ) = 1 {\displaystyle \lim _{b\to \infty }\int \limits _{1}^{b}{\frac {dx}{x^{2}}}=\lim _{b\to \infty }\left({\frac {1}{1}}-{\frac {1}{b}}\right)=\lim _{b\to \infty }\left(1-{\frac {1}{b}}\right)=1} This improper integral can be interpreted as the area of the unbounded region between f ( x ) = 1 x 2 {\displaystyle f(x)={\frac {1}{x^{2}}}} , y = 0 {\displaystyle y=0} (the x {\displaystyle x} -axis), and x = 1 {\displaystyle x=1} . 1. Suppose ∫ a b f ( x ) d x {\displaystyle \int \limits _{a}^{b}f(x)dx} exists for all b ≥ a {\displaystyle b\geq a} . Then we define ∫ a ∞ f ( x ) d x = lim b → ∞ ∫ a b f ( x ) d x {\displaystyle \int \limits _{a}^{\infty }f(x)dx=\lim _{b\to \infty }\int \limits _{a}^{b}f(x)dx} , as long as this limit exists and is finite. If it does exist we say the integral is convergent and otherwise we say it is divergent. 2. Similarly if ∫ a b f ( x ) d x {\displaystyle \int \limits _{a}^{b}f(x)dx} exists for all a ≤ b {\displaystyle a\leq b} we define ∫ − ∞ b f ( x ) d x = lim a → − ∞ ∫ a b f ( x ) d x {\displaystyle \int \limits _{-\infty }^{b}f(x)dx=\lim _{a\to -\infty }\int \limits _{a}^{b}f(x)dx} 3. Finally suppose c {\displaystyle c} is a fixed real number and that ∫ − ∞ c f ( x ) d x {\displaystyle \int \limits _{-\infty }^{c}f(x)dx} and ∫ c ∞ f ( x ) d x {\displaystyle \int \limits _{c}^{\infty }f(x)dx} are both convergent. Then we define ∫ − ∞ ∞ f ( x ) d x = ∫ − ∞ c f ( x ) d x + ∫ c ∞ f ( x ) d x {\displaystyle \int \limits _{-\infty }^{\infty }f(x)dx=\int \limits _{-\infty }^{c}f(x)dx+\int \limits _{c}^{\infty }f(x)dx} First we give a definition for the integral of functions which have a discontinuity at one point. If f {\displaystyle f} is continuous on the interval [ a , b ) {\displaystyle [a,b)} and is discontinuous at b {\displaystyle b} , we define ∫ a b f ( x ) d x = lim c → b − ∫ a c f ( x ) d x {\displaystyle \int \limits _{a}^{b}f(x)dx=\lim _{c\to b^{-}}\int \limits _{a}^{c}f(x)dx} If the limit in question exists we say the integral converges and otherwise we say it diverges. Similarly if f {\displaystyle f} is continuous on the interval ( a , b ] {\displaystyle (a,b]} and is discontinuous at a {\displaystyle a} , we define ∫ a b f ( x ) d x = lim c → a + ∫ c b f ( x ) d x {\displaystyle \int \limits _{a}^{b}f(x)dx=\lim _{c\to a^{+}}\int \limits _{c}^{b}f(x)dx} Finally suppose f {\displaystyle f} has an discontinuity at a point c ∈ ( a , b ) {\displaystyle c\in (a,b)} and is continuous at all other points in [ a , b ] {\displaystyle [a,b]} . If ∫ a c f ( x ) d x {\displaystyle \int \limits _{a}^{c}f(x)dx} and ∫ c b f ( x ) d x {\displaystyle \int \limits _{c}^{b}f(x)dx} converge we define ∫ a b f ( x ) d x {\displaystyle \int \limits _{a}^{b}f(x)dx} = ∫ a c f ( x ) d x + ∫ c b f ( x ) d x {\displaystyle \int \limits _{a}^{c}f(x)dx+\int \limits _{c}^{b}f(x)dx} We can also give a definition of the integral of a function with a finite number of discontinuities. Suppose f {\displaystyle f} is continuous on [ a , b ] {\displaystyle [a,b]} except at points c 1 < c 2 < ... < c n {\displaystyle c_{1}<c_{2}<\ldots <c_{n}} in [ a , b ] {\displaystyle [a,b]} . We define ∫ a b f ( x ) d x = ∫ a c 1 f ( x ) d x + ∫ c 1 c 2 f ( x ) d x + ∫ c 2 c 3 f ( x ) d x + ⋯ + ∫ c n − 1 c n f ( x ) d x + ∫ c n b f ( x ) d x {\displaystyle {\begin{aligned}\int \limits _{a}^{b}f(x)dx=\int \limits _{a}^{c_{1}}f(x)dx+\int \limits _{c_{1}}^{c_{2}}f(x)dx+\int \limits _{c_{2}}^{c_{3}}f(x)dx+\cdots +\int \limits _{c_{n-1}}^{c_{n}}f(x)dx+\int \limits _{c_{n}}^{b}f(x)dx\end{aligned}}} as long as each integral on the right converges. Notice that by combining this definition with the definition for improper integrals with infinite endpoints, we can define the integral of a function with a finite number of discontinuities with one or more infinite endpoints. There are integrals which cannot easily be evaluated. However it may still be possible to show they are convergent by comparing them to an integral we already know converges. Theorem (Comparison Test) Let f , g {\displaystyle f,g} be continuous functions defined for all x ≥ a {\displaystyle x\geq a} . Suppose g ( x ) ≥ f ( x ) ≥ 0 {\displaystyle g(x)\geq f(x)\geq 0} for all x ≥ a {\displaystyle x\geq a} . Then if ∫ a ∞ g ( x ) d x {\displaystyle \int \limits _{a}^{\infty }g(x)dx} converges so does ∫ a ∞ f ( x ) d x {\displaystyle \int \limits _{a}^{\infty }f(x)dx} . Suppose f ( x ) ≥ h ( x ) ≥ 0 {\displaystyle f(x)\geq h(x)\geq 0} for all x ≥ a {\displaystyle x\geq a} . Then if ∫ a ∞ h ( x ) d x {\displaystyle \int \limits _{a}^{\infty }h(x)dx} diverges so does ∫ a ∞ f ( x ) d x {\displaystyle \int \limits _{a}^{\infty }f(x)dx} . A similar theorem holds for improper integrals of the form ∫ − ∞ b f ( x ) d x {\displaystyle \int \limits _{-\infty }^{b}f(x)dx} and for improper integrals with discontinuities. To apply the comparison theorem you do not really need g ( x ) ≥ f ( x ) ≥ 0 {\displaystyle g(x)\geq f(x)\geq 0} for all x ≥ a {\displaystyle x\geq a} . What we actually need is this inequality holds for sufficiently large x {\displaystyle x} (i.e. there is a number c {\displaystyle c} such that g ( x ) ≥ f ( x ) {\displaystyle g(x)\geq f(x)} for all x ≥ c {\displaystyle x\geq c} ). For then ∫ a ∞ f ( x ) d x = ∫ a c f ( x ) d x + ∫ c ∞ f ( x ) d x {\displaystyle \int \limits _{a}^{\infty }f(x)dx=\int \limits _{a}^{c}f(x)dx+\int \limits _{c}^{\infty }f(x)dx} so the first integral converges if and only if third does, and we can apply the comparison theorem to the ∫ c ∞ f ( x ) d x {\displaystyle \int \limits _{c}^{\infty }f(x)dx} piece.
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