CA1 MATHS

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  TITLE IMPROPER INTEGRATION
• SUBJECT NAME MATHEMATICS-II
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  SUBJECT CODE BSM-102
• PRESENTED BY
  ISHAN NAG
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  UNIVERSITY ROLL NO. 14200323021
  
• DEPARTMENT OF
  ELECTRONICS AND COMMUNICATION ENGINEERING

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• CONTENTS
• INTRODUCTION
  
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• TECHNIQUES OF IMPROPER INTEGRALS
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• APPLICATIONS OF IMPROPER INTEGRALS
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• CONCLUSION
  
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• REFERENCES 9
• ACKNOWLEDGEMENT
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INTRODUCTION In the vast landscape of
calculus, when standard methods face limitations,
improper integrals play an important role. The
definite integral, which is good at calculating
the area under a curve in a finite interval,
matches its own game with infinite functions or
discontinuities. The improper integral came on
the scene as a solution to these
complexities. Type I improper integral, where
the limit of integration extends to infinity, and
Type II, which deals with discontinuities in the
integration interval, form the basis. Techniques
such as integration by parts, trigonometric
substitution, and dividing fractions become
important tools for solving these
complexities. In addition to theoretical
constructions, the improper integral finds
real-world applications in a variety of
fieldsdetermining the convergence of infinite
series, calculating the area under a curve which
extends to infinity, and contributing to
solutions in physics, engineering, and
economics. As we conduct these research
projects, we learn nuances, perform convergence
tests, and witness the transformative impact of
improper integrals on mathematical and scientific
research. This journey promises intellectual
satisfaction and a deep appreciation for adapting
mathematical concepts when faced with the
complexities of the real world.



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• TECHNIQUES OF IMPROPER INTEGRALS
• Improper integrals when solving functions with
  infinite domains or singularities require a
  thorough understanding of evaluation methods.
  Here, we review the basic methodology covering
  integral types and show that the improper
  integral is a form of the definite integral.
• DEFINITE INTEGRAL
• The definite integral, symbolized as, ?? ?? ??
  ?? ???? is defined as the net area under the
  curve f(x) between two points a, b on the X-axis.
  The Definite Integral represents the cumulative
  effect of a specific function in a certain
  interval.
• Types of Definite Integrals
• Proper Definite Integral Standard definite
  integral over a closed, finite interval a, b.
  It calculates the net area under the curve of the
  integrand within this fixed and finite range.
• Improper Definite Integral Involves integrating
  over intervals with one or both bounds extending
  to infinity or with singularities in the
  integrand. Evaluation often requires extra
  techniques like limits to handle unbounded or
  singular behaviour.
• Types of Improper Integrals
• Type-I Improper Integral This type of integral
  includes an integrand which has one or both
  limits of integration extending to infinity.
  Example 1 8 ( 1 ?? 2 )????
• Type-II Improper Integrals This type of integral
  includes an integrand which becomes infinite or
  discontinuous within the interval of integration.
  Example 0 1 ( 1 v?? )????

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• Comparison Test
• Comparison with a simple known integral This
  technique involves comparison of a provided
  improper integral with a known integral of a
  simple function. By utilizing the behaviour of
  known integrals, we can gain insight into the
  convergence or divergence of the original
  integral.
• Useful for integrands lacking a direct
  antiderivative In scenarios where finding the
  antiderivative is quite tedious, comparison with
  known integrals provides a practical approach to
  find out the convergence with much ease.
• Integration By Parts
• Use ? udv uv- ? vdu formula to simplify
  Integration by parts is a process for simplifying
  integrals of products of functions. It involves
  splitting the integral into two parts and using
  the integration-by-parts formula that relates the
  integral of the product to the product of
  functions minus another integral.
• Requirement of a strategic choice of u and dv
  The choice of u and dv plays an important role
  here. The u is chosen based on its ease of
  differentiation and dv is chosen based on its
  ease of integration.
• Partial decomposition of the fraction
• Decompose rational functions into simpler
  fractions This is used for integrals which
  involves rational functions. This involves
  representing a complex fraction as a sum of
  simple fractions with unknown coefficients.
• Solving unknown coefficients Uncertain
  coefficients are determined by equating the
  decomposed expression with the original logic
  function and solving the resulting system of
  equations.
• Trigonometric Substitution
• Use trigonometric identities for simplification
  Trigonometric substitution is employed to
  simplify integrals involving square roots or
  quadratic expressions. It involves substituting
  trigonometric functions to transform the integral
  into a more manageable form.
• Selection of substitutions based on the form of
  integral The choice of the trigonometric
  substitutions totally depends on the integral's
  form. The common substitutions involve
  expressions like a2-x2 or a2x2

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• Breakdown of integrals in singularities
• Carefully address the individual points When the
  integrals include singular points or points of
  discontinuity,
• dividing the integral into smaller intervals
  around these points allows for a more detailed
  analysis and mor ease.
• Break the integral into smaller intervals The
  integral is divided into the intervals around the
  singular points and the behaviour of
• the function is analysed separately in each of
  intervals.
• Absolute convergence
•  Analyse the absolute value convergence Absolute
  convergence involves the study of the convergence
  behaviour of the absolute value
• of the integrand rather than the integrand
  itself. This can simplify the evaluation process.
•  Especially useful for oscillatory functions or
  alternating series In cases where the integrand
  oscillates or involve
• s alternating signs, absolute convergence helps
  to focus on the overall behaviour without being
  affected by sign changes.

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APPLICATIONS OF IMPROPER INTEGRALS Physics
Improper integrals play an important role in
the field of Physics. They are essential for
determination of the mass of objects which have
irregular density distributions and provide
better understanding of the physical properties
of complex structures. Improper integrals are
helpful in calculating the center of mass, which
offers insight into the equilibrium and stability
of physical systems. Improper integrals model
radioactive decay and provide insights into
half-life and decay processes that are key to
nuclear physics research. Probability and
Statistics In the field of probability and
statistics, the improper integral is important
for computation of the probabilities of
continuous random variables and for determining
statistical moments, providing fundamental tools
for data analysis and modeling. Engineering
Improper integrals are an important part of
signal processing, specifically while dealing
with signals of infinite duration or bandwidth.
And plays an important role in the analysis and
manipulation of signals for applications in
communication, control systems, and image
processing. Finance and Economics Improper
Integrals can also be primary guides for making
decisions. The net present value (NPV),
profitability of investments, and
discounted-cash flow forecasts for future
evaluations and planning of long-term goals are
calculated through them. Electrical and
Computer Engineering In algorithmic analysis,
improper integration is used to evaluate the
efficiency of an algorithm, especially when the
size of the input reaches infinity. They play an
important role in understanding and improving
computational processes. Biology and Medicine
Improper integrals are important in
pharmaceutical field , drug absorption modeling
and enzyme kinetics. It offers a quantitative
understanding of drug effects, guiding the
development of precise drugs with minimal side
effects. This mathematical approach influences
pharmacology and advances drug science to ensure
optimal drug delivery.
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CONCLUSION In summary, the use of improper
integrals in physics, statistics, engineering,
finance, and elsewhere has demonstrated their
usefulness in solving complex real-world
problems. The versatility of mathematical
methods not only deepens our understanding of
fundamental processes, but also takes progress in
various scientific and technological fields. When
we examine the complexity of this application,
it becomes clear that the imprecise integral
serves as an indispensable tool, providing
invaluable insight and solutions in various areas.
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• REFERENCES
• Larson, R., Edwards, B. (2009). "Calculus of
  a Single Variable." Cengage Learning.
• Thomas Jr, G. B., Weir, M., Hass, J.,
  Giordano, F. R. (2018). "Thomas' Calculus Early
  Transcendentals." Pearson.
• Stewart, J. (2015). "Calculus Early
  Transcendentals." Cengage Learning.
• Simmons, G. F. (1996). "Introduction to Topology
  and Modern Analysis." McGraw-Hill.
• Kreyszig, E. (2011). "Advanced Engineering
  Mathematics." John Wiley Sons.
• Wikipedia (https//en.wikipedia.org/wiki/Im
  proper_integral)
• Khan Academy ( https//www.khanacademy.org/
  )

ACKNOWLEDGEMENT I extend my heartfelt
appreciation to Dr. S.K. Mosiur Rahaman and Dr.
Bikash Chandra Bhui for their guidance and
mentorship in the development of this
presentation. Their combined expertise and
encouragement have enriched the content, ensuring
a thorough exploration of the applications of
improper integrals along with its techniques and
applications. I am grateful for their commitment
to academic excellence and their invaluable
contributions to my learning journey.
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