GATE Question on Integration by Parts Method

Duration: 7 min

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Module outline

  1. General Intelligence & Reasoning: Analogy, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Series (Number and Letter Series) (Numerical Relations and Reasoning), Coding Decoding, Blood Relations, Directions (Direction Test), Ranking, Order and Sequence, Logical Sequence Of Words, Syllogisms, Venn Diagram, Deductive and Inductive Reasoning (Logical Deduction and Induction) (Prepositional Reasoning), Selection Decision Table (Decision Making), Data Sufficiency, Puzzle Solving (Puzzles), Data Arrangement, Cubes & Dices, Calendar, Clock, Arithmetic, Number System, HCF LCM, Verbal Analogy (Word Based Analogy), Visual Recognition and Figure Completion, Paper Folding and Cutting, Transformation of Shapes, Clocks and Calendars, Direction Sense, Cubes and Dice, Patterns in Two-Dimensional Figures, Seating Arrangements, Analogy and Classification, Logical Deduction and Induction, Alphabet and Word Tests, Mathematical Operations, Series Completion, Simplification and Approximation, Coding and Decoding, Numerical Reasoning, Assembling and Grouping of Shapes, Three-Dimensional Patterns and Visualization, Vocabulary, Number System, Averages, Mixtures and Alligation, Simple and Compound Interest, Percentages, Algebra, Speed, Time and Distance, Seating and Arrangement Puzzles, Traverse Surveying, Order, Ranking and Sequence, Constraint Puzzles, Syllogisms and Venn Diagrams, Reading Comprehension, Critical Reasoning, Permutations and Combinations, Narrative Sequencing, Project Monitoring, Resources and Safety, Profit, Loss and Discount, Ratio and Proportion, Time and Work, Mensuration, Data Interpretation, Powers, Surds and Logarithms, Sequences and Series, Articles and Determiners, Sentence Completion, Data Sufficiency, Blood Relations, Simple Interest and Compound Interest, Mixture and Alligation, Sequence and Series (Series), Speed Time and Distance, Algebra, Profit and Loss, Powers and Exponents (Surds and Indices), Permutation and Combination
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  3. Engineering Mathematics: Set Theory, Relations, Functions, Graph Theory, Group Theory, Propositional and Predicate Logic, Linear Algebra, Calculus, Probability, Statistics, Signal Space and Optimum Detection, Probability, Statistics
  4. Digital Logic: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Number System, Number Representation, Floating Point Rep, Number Systems and Codes, Registers and Counters, Boolean Algebra and Logic Gates, Microprocessors and Microcontrollers, Logic Families and CMOS Logic, Microprocessors, Logic Gates and Boolean Algebra, Minimization of Boolean Functions, Timing, Hazards and Testing, Logic Families, Combinational Logic Design, Arithmetic Circuits, Waveform Generators and Timer Circuits, Semiconductor Memories and Programmable Logic Devices, Combinational Logic Circuits, Latches and Flip-Flops, Finite State Machines, Sequential Circuits and State Machines, Minimization of Boolean Functions, Arithmetic Circuits, Latches and Flip-Flops, Number Systems and Codes, Registers and Counters
  5. Computer Organisation & Architecture: Basics of COA, Cache Memory Organization, Input Output Organisation, Pipelining, Instr Formats & Modes, Control Unit Design, Programming the Basic Computer, Register Transfer and Microoperations, Semiconductor Memories and Programmable Logic, Processor Organization and Instructions, Memory Hierarchy and Input/Output, Registers and Counters
  6. Programming & Data Structures: C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling, Introduction to DS, Array, Stack, Queue, Linked List, Tree, Graphs, Hashing, HTML Basics, OOPS with C++, Java, Object Oriented Technology
  7. Databases: Basics of DBMS, ER Diagram, Relational Model & Functional Dependencies, Keys & Integrity Constraints, Normalization (1NF - BCNF), Decomposition Properties & 4NF, File Organization & Indexing, Relational Algebra, SQL, Relational Calculus, Transaction Management, Concurrency Control, Database Recovery, Database Security & Authorization
  8. Algorithms: Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Dynamic Programming, Minimum Spanning Trees, Shortest Path Algos
  9. Theory of Computation: Introduction to TOC, Deterministic FA (DFA), Non-Deterministic FA, Regular Expressions, Grammar, Regular Language Properties, Moore & Mealy Machines, Pushdown Automata & CFG, Turing Machines, Complexity Theory
  10. Compiler Design: Intro to Compilers, Lexical Analysis, Grammar & CFG, Syntax Analysis: Top-Down, Syntax Analysis: Bottom-Up, Semantic Analysis & SDT, Intermediate Code Gen, Code Optimization, Run Time Environment
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  13. Full Length Mock Tests:
  14. Sectional Tests:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This educational video features a calculus lecture by Yash Jain Sir, focusing on solving a definite integral problem from the GATE examination. The core task is to evaluate the integral of x squared times cosine x from 0 to pi. The instructor systematically applies the integration by parts technique, utilizing the LIATE rule to determine the choice of u and dv. The lesson covers setting up the integral, evaluating boundary terms, and handling a secondary integral that arises from the first application of the method.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video begins with an intro screen displaying CALCULUS surrounded by various mathematical formulas. The instructor introduces the problem integral from 0 to pi of x squared cos x dx written on the whiteboard. He lists essential trigonometric values needed for evaluation: cos 0 equals 1, cos pi equals negative 1, sin 0 equals 0, and sin pi equals 0. He then introduces the LIATE rule Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential to guide the selection of parts for integration by parts. He identifies x squared as the Algebraic part u and cos x as the Trigonometric part dv. He writes the standard integration by parts formula integral uv equals u integral v minus integral u prime integral v.

  2. 2:00 – 5:00 02:00-05:00

    The instructor proceeds to apply the formula. He calculates the integral of cos x as sin x. Substituting these into the formula, he writes bracket x squared sin x bracket from 0 to pi minus integral from 0 to pi of 2x sin x dx. He evaluates the first term bracket x squared sin x bracket from 0 to pi by substituting the limits, noting that sin pi and sin 0 are both zero, causing this entire term to vanish. This leaves the remaining integral negative integral from 0 to pi of 2x sin x dx. He factors out the constant 2 to get negative 2 integral from 0 to pi of x sin x dx. He prepares to apply integration by parts again to this new integral, identifying x as u and sin x as dv. He writes the integral of sin x as negative cos x.

  3. 5:00 – 6:51 05:00-06:51

    The instructor completes the second integration by parts. He writes the expression negative 2 bracket x negative cos x minus integral 1 negative cos x bracket. He simplifies the inner integral to integral cos x, which evaluates to sin x. The full expression becomes negative 2 bracket negative x cos x plus sin x bracket from 0 to pi. He evaluates the limits at pi and 0. At the upper limit pi, the term becomes negative 2 bracket negative pi cos pi plus sin pi bracket, which simplifies to negative 2 bracket negative pi negative 1 plus 0 equals negative 2 pi. At the lower limit 0, the term is zero. The final result is negative 2 pi. The video concludes with a THANKS FOR WATCHING screen.

The lecture effectively demonstrates a multi-step integration problem. By breaking down the process into two distinct applications of integration by parts, the instructor shows how to reduce the power of the polynomial term x squared until it disappears. The use of the LIATE rule ensures the correct choice of functions, preventing unnecessary complexity. The final evaluation relies heavily on the specific values of trigonometric functions at the boundaries 0 and pi.

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