Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate → Calculus → Integral Calculus ↓


Improper Integrals


In calculus, integration is a fundamental concept that helps us find areas under curves, among many other applications. This concept gets a little more complicated when we move into the realm of improper integrals. An improper integral is an integral that arises when the function we want to integrate is unbounded or when the limits of integration are infinite, which are situations that do not fit the typical setup of a definite integral.

Broadly speaking, improper integrals help us extend the ideas of definite integrals to a wider class of functions and intervals that includes infinity. They are an essential topic in undergraduate mathematics, as they involve integrals that are frequently encountered when dealing with real-world problems.

Types of improper integrals

There are two primary types of improper integrals:

  1. Improper integrals with infinite limits
  2. Improper integrals with unbounded integrals

1. Improper integrals with infinite limits

The first type occurs when one or both of the limits of integration are -∞ or ∞. Let's consider the integral:

    ∫ a ∞ f(x) dx

Here, the upper limit of integration is infinity. We need to evaluate it as a limit. We rewrite it as:

    lim b→∞ ∫ a b f(x) dx

If this limit exists and is finite, we say that the improper integral converges. If it does not converge, we say that it diverges.

Let us consider a simple example:

    ∫ 1 ∞ (1/x 2 ) dx

Rewriting this using limits, we get:

    lim b→∞ ∫ 1 b (1/x 2 ) dx

Finding the integral, we get:

    = lim b→∞ [-1/x] 1 b

= lim b→∞ [-1/b + 1/1]

= lim b→∞ (1 - 1/b)

= 1

Hence this integral converges to 1.

2. Improper integrals with unbounded integrals

The second type of improper integral arises when the integrand has an infinite discontinuity within the interval [a, b] or at its boundaries. For example:

    ∫ 0 1 (1/√x) dx

Here, the integral becomes infinite as x approaches zero. We have to write the integral as follows:

    lim t→0⁺ ∫ t 1 (1/√x) dx

Let us evaluate it step by step:

    = lim t→0⁺ [2√x] t 1

= lim t→0⁺ [2√1 - 2√t]

= 2 - 0

= 2

In this case the integral converges to 2.

Visual example

To better understand these ideas, let's use some visual examples.

Imagine that we are integrating the function f(x) = 1/x 2 over the interval [1, ∞]. This function looks like this:

f(x) = 1/x² 1 X Y

This graph shows that as x approaches infinity, the value of the function decreases and approaches the x-axis, which indicates convergence in this context.

Now, consider f(x) = 1/√x on the interval [0, 1]. The function looks like this:

f(x) = 1/√x 0 1 Y

This shows that the function rises to infinity as it approaches zero, but falls back towards 1 as it moves to the right.

Testing the convergence of improper integrals

It is important to determine the convergence or divergence of improper integrals. There are several methods for testing convergence:

1. Comparison test

In the comparison test the improper integral is compared with another integral whose convergence is already known. The basic idea is this:

  • If 0 ≤ f(x) ≤ g(x) for all x in the interval, and if ∫ g(x) dx is convergent, then ∫ f(x) dx is also convergent.
  • If f(x) ≥ g(x) ≥ 0 for all x, and if ∫ g(x) dx diverges, then ∫ f(x) dx also diverges.

Example: Show that ∫ 1 ∞ (1/(x²+1)) dx converges.

Compare this with ∫ 1 ∞ (1/x²) dx, which we know is convergent. Clearly, 0 ≤ 1/(x²+1) ≤ 1/x² Thus, by the comparison test, ∫ 1 ∞ (1/(x²+1)) dx is also convergent.

2. Limit comparison test

The limit comparison test is an extension of the comparison test, where we assume that f(x) > 0 and g(x) > 0.

define:

    lim x→∞ [f(x)/g(x)] = l

If L is a positive finite number, then ∫ f(x) dx and ∫ g(x) dx will both either converge or diverge.

Example: Consider ∫ 1 ∞ (3/(2x²+5x+3)) dx.

Let g(x) = 1/x². Now calculate:

    L = lim x→∞ [(3/(2x²+5x+3))/(1/x²)]

Simplification:

    = lim x→∞ [3x²/(2x²+5x+3)]

= lim x→∞ [3/(2+(5/x)+(3/x²))]

= 3/2

Since L is a positive finite number, both integrals converge simultaneously. Therefore, ∫ 1 ∞ (3/(2x²+5x+3)) dx converges.

Applications and significance

Improper integrals are found everywhere in mathematics and science. They are important in probability theory, especially in determining distributions with infinite support. They also occur in physics when evaluating the total work done by a force when the distance is infinite.

In addition, many results and techniques in calculus, such as the residue theorem in complex analysis, beautiful proofs in geometry, and the solutions of differential equations, also use improper integrals.

Conclusion

Understanding improper integrals is an integral part of a strong knowledge of calculus. Their convergence or divergence determines whether certain regions, probabilities, and physical quantities can be effectively calculated. Mastering this topic requires being comfortable with limits and familiar with various tests for convergence, which paves the way for deeper mathematical insights.

It's important to work through a number of examples, both analytically and graphically, to solidify these concepts and develop an intuitive understanding that you can apply in advanced mathematical contexts.


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