Improper Integrals Improper Integrals
Introduction
Proper integrals are bounded to a finite domain with a finite function. For example:
\int^{30}_1{\ln(x) \, dx}
Improper integrals are either on an unbounded (infinite) domain or have an unbounded function.
For example,
\int^{\infty}_1{\ln(x) \, dx}
On an unbounded domain.
\int^{3}_0{\frac{1}{x} \, dx}
Has an unbounded function.
Unbounded Domains
\begin{align*}
\int^{\infty}_1{\ln(x) \, dx} &= x\ln|x| + \ln|x|\Big|_1^{\infty} \\ \text{} \\
&= \infty\ln|\infty| + \ln|\infty| - (\ln|1| + \ln|1|)
\end{align*}
This is not a valid solution. We must change the integral.
Antiderivatives, by their nature, are differentiable. Therefore, we can evaluate the integral as a limit.
\begin{align*}
\int^{\infty}_1{\ln(x) \, dx} &= \lim_{n \to \infty} \int^{n}_1{\ln(x) \, dx} \\ \text{} \\
&= \lim_{n \to \infty} x\ln|x| + \ln|x| \Big|^n_1 \\ \text{} \\
&= \lim_{n \to \infty} n\ln|n| + \ln|n| - (\ln|1| + \ln|1|) \\ \text{} \\
&= \infty
\end{align*}
Example #2:
\begin{align*}
\int^{\infty}_5{\frac{1}{x^2} \, dx} &= \lim_{n \to \infty} \int^{n}_5{\frac{1}{x^2} \, dx} \\ \text{} \\
&= \lim_{n \to \infty} -\frac{1}{x} \Big|^n_{5} \\ \text{} \\
&= \lim_{n \to \infty} -\frac{1}{n} + \frac{1}{5} \\ \text{} \\
&= \frac{1}{5}
\end{align*}
Unbounded Functions
\begin{align*}
\int_0^3 \frac{1}{x} \, dx &= \ln|x|\Big|_0^3 \\ \text{} \\
&= \ln|3| - \ln|0|
\end{align*}
Since
\ln|0|
does not exist, we cannot solve this integral. We must also evaluate this integral as a limit.
\begin{align*}
\int_0^3 \frac{1}{x} \, dx &= \lim_{n \to 0^+} \int^{3}_n{\frac{1}{x} \, dx} \\\\
&= \lim_{n \to 0^+} \ln|x| \Big|^3_n \\ \text{} \\
&= \lim_{n \to 0^+} \ln|3| - \ln|n| \\ \text{} \\
&= \ln|3| + \infty \\ \text{} \\
&= \infty
\end{align*}
Here is another example:
\begin{align*}
\int_0^7 \frac{1}{x^2} \, dx &= \lim_{n \to 0^+} \int^{7}_n{\frac{1}{x^2} \, dx} \\\\
&= \lim_{n \to 0^+} -\frac{1}{x} \Big|^7_n \\ \text{} \\
&= \lim_{n \to 0^+} -\frac{1}{7} + \frac{1}{n} \\ \text{} \\
&= -\frac{1}{7} + \infty \\ \text{} \\
&= \infty
\end{align*}