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Improper integration.

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Recall

A definite integral is a mathematical concept used to find the area under a curve between two fixed limits. It is represented as $\int_{a}^{b}f(x)dx$, where a and b are the lower and upper limits, and f(x) is the integrand.

A definite integral of a function is defined as the area of the region bounded by the function's graph or curve between two points, say a and b, and the x-axis. The integral of a real-valued function f(x) on an interval [a, b] is written or expressed as $\int_{a}^{b} f(x)dx$ and it is illustrated in Figure 1.a.  

The Fundamental Theorem of Calculus states roughly that the integral of a function f over an interval is equal to the change of any antiderivate F (F'(x) = f(x)) between the ends of the interval, i.e., $\int_{a}^{b} f(x)dx = F(b)-F(a)=F(x) \bigg|_{a}^{b}$

 

Improper integration

Definite integrals have specific limits of integration which determine the net area under the curve of the function between these limits. Improper integrals arise when the interval of integration is unbounded, that is, $\int_{a}^{∞} f(x)dx = \lim_{b \to ∞}\int_{a}^{b} f(x)dx$, or when the integrand is undefined at certain points within the interval of integration. Techniques for evaluating improper integrals involve taking limits as the integration bounds approach infinity or as singularities are approached.

An improper integral converges if the limit defining it exists, that is, the total area [a, ∞) under the curve f(x) is finite. In other words, improper integrals are said to be convergent if the limit is finite and that limit is the value of the improper integral.

An improper integral diverges if the limit defining it does not exist It is also possible for an improper integral to diverge to infinity. In that case, one may assign the value of ∞ (or −∞) to the integral. However, other improper integrals may simply diverge in no particular direction. This is called divergence by oscillation.

Solved exercises

$\int_{-∞}^{0} \frac{1}{1+x^2}dx = \lim_{a \to -∞} \int_{a}^{0} \frac{1}{1+x^2}dx = \lim_{a \to -∞} arctan(x)\bigg|_{a}^{0} = $

$lim_{a \to -∞} arctan(0)-arctan(a) = lim_{a \to -∞} 0-arctan(a) = \frac{π}{2} ⇒$[By symmetry] $\int_{-∞}^{∞} \frac{1}{1+x^2}dx =\frac{π}{2}+\frac{π}{2} = π.$

$\lim_{b \to ∞} (-\frac{ln(b)}{b}-\frac{1}{b}) - (-\frac{ln(1)}{1}-\frac{1}{1}) = \lim_{b \to ∞} (-\frac{ln(b)}{b}-\frac{1}{b})-(-0-1)=0-(0-1)=1.$

By L’Hospital Rule, $\lim_{b \to ∞} (\frac{ln(b)}{b}) = \lim_{b \to ∞} \frac{\frac{1}{b}}{1} = \lim_{b \to ∞} \frac{1}{b} = 0$

$\lim_{b \to 2⁻} ln|b^2-4| -ln(4)=-∞,$ so the integral diverges.

$= \frac{2}{3} \lim_{b \to ∞} (\sqrt{b}-{\sqrt{4}})= \frac{2}{3}(∞ -2) = ∞.$

Therefore, $\int_{0}^{∞} e^{-kx}dx =\lim_{b \to ∞}\frac{-1}{k}e^{-kb}-(\frac{-1}{k}) = 0+-\frac{-1}{k} = \frac{1}{k}$.

$\lim_{b \to ∞} lnb -ln1 = \lim_{b \to ∞} lnb = ∞$, so our original integral diverges.

However, this is equal to $= \lim_{b \to ∞} (-cos(b)+cos(0)) = \lim_{b \to ∞} -cos(b)+1$, this limit does not exist, this improper integral is divergent.

$ = \lim_{a \to -∞} (e^0(0-1))-(e^a(a-1)) = \lim_{a \to -∞} -1-ae^a+e^a = -1 -\lim_{a \to -∞} ae^a +0$. However, $\lim_{a \to -∞} ae^a = \lim_{a \to -∞} \frac{a}{e^{-a}}$[L’Hôpital’s, -∞/∞] $\lim_{a \to -∞} \frac{1}{-e^{-a}} = 0$, hence $\int_{-∞}^{0} xe^xdx = -1 -0 + 0 = -1.$

$= \lim_{b \to ∞} \frac{b^{-p+1}}{-p+1}-\frac{1^{-p+1}}{-p+1}$. Therefore, there are three cases:

  1. p = 1, $\int_{1}^{∞} \frac{dx}{x} = ∞$, it diverges (it was solved previously).
  2. p < 1, $\int_{1}^{∞} \frac{dx}{x^p} = ∞$, it diverges.
  3. p > 1, $\int_{1}^{∞} \frac{dx}{x^p} = -\frac{1}{-p+1} = \frac{1}{p-1}$, so it converges.

$= \frac{1}{2}·\lim_{b \to ∞}(tan^{-1}(b)-tan^{-1}(0)) = \frac{1}{2}·\lim_{b \to ∞}(tan^{-1}(b)-0) = \frac{1}{2}·tan^{-1}(∞) = \frac{1}{2}·\frac{π}{2} = \frac{π}{4}.$

Bibliography

This content is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License and is based on MIT OpenCourseWare [18.01 Single Variable Calculus, Fall 2007].
  1. NPTEL-NOC IITM, Introduction to Galois Theory.
  2. Algebra, Second Edition, by Michael Artin.
  3. LibreTexts, Calculus. Abstract and Geometric Algebra, Abstract Algebra: Theory and Applications (Judson).
  4. Field and Galois Theory, by Patrick Morandi. Springer.
  5. Michael Penn, and MathMajor.
  6. Contemporary Abstract Algebra, Joseph, A. Gallian.
  7. YouTube’s Andrew Misseldine: Calculus. College Algebra and Abstract Algebra.
  8. MIT OpenCourseWare 18.01 Single Variable Calculus, Fall 2007 and 18.02 Multivariable Calculus, Fall 2007.
  9. Calculus Early Transcendentals: Differential & Multi-Variable Calculus for Social Sciences.
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