Solving Integration: Integration by Parts Method

Duration: 10 min

This video lesson is available to enrolled students.

Return to /learn/ISRO/general-aptitude-reasoning/calculus/integration/asset-solving-integration-integration-by-parts-method after enrolling

Inside: a video lesson and guided study material.

Module outline

  1. About Course: About the Course
  2. General Aptitude & Reasoning: Number System, Percentage, Ratio and Proportion (Ratios), Average, Profit and Loss, Simple Interest and Compound Interest, Time and Work, Speed Time and Distance, Mixture and Alligation, Data Interpretation, Algebra, Probability, Permutation and Combination, Venn Diagram, HCF LCM, Divisibility Rules, Age Problems, Pipe and Cistern, Partnership, Linear Algebra, Sequence and Series (Series), Statistics, Powers and Exponents (Surds and Indices), Arithmetic, Mensuration and Geometry, Calculus, Trigonometry, Vectors, Speed Mathematics and Basic Mathematics, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Analogy, Series (Number and Letter Series) (Numerical Relations and Reasoning), Puzzle Solving (Puzzles), Syllogisms, Deductive and Inductive Reasoning (Logical Deduction and Induction) (Prepositional Reasoning), Calendar, Seating Arrangements, Coding Decoding, Blood Relations, Directions (Direction Test), Clock, Analytical Reasoning (Counting Figures Reasoning), Ranking, Order and Sequence, Logical Sequence Of Words, Inequality, Data Sufficiency, Cubes & Dices
  3. Computer Networks & Cybersecurity: Introduction to CN, Data Communication, DLL: Access Control, DLL: Flow Control, DLL: Error Control, DLL: Framing, Data Link Layer - Ethernet, Net Layer: IPv4 & Proto, Net Layer: IP Addressing, Net Layer:Routing Protocol, Transport Layer Services, TL: Congestion & UDP, Application Layer, Hardware Basics, Network Security
  4. Programming, Data Structures & Algorithms: C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling, OOPS with C++, Introduction to DS, Array, Stack, Queue, Linked List, Tree, Graphs, Hashing, Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Dynamic Programming, Minimum Spanning Trees, Shortest Path Algos, Advanced Algorithms
  5. AI, Machine Learning, Big Data & Cloud: Emerging Tech Trends, Artificial Intelligence, Big Data Systems, Approaches to AI, Search Algorithms, Knowledge Representation, Cloud Computing
  6. Digital Logic & Computer Organization: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Number System, Number Representation, Basics of COA, Floating Point Rep, Cache Memory Organization, Input Output Organisation, Pipelining, Instr Formats & Modes, Register Transfer and Microoperations, Control Unit Design, Programming the Basic Computer
  7. TOC & Compiler: 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, 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
  8. Discrete Mathematics: Set Theory, Relations, Functions, Graph Theory, Group Theory, Propositional and Predicate Logic
  9. Operating System: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Threads & Process Creation, Deadlock, Memory Management, Virtual Memory, Disc Scheduling, File Management, Unix OS, Linux OS, Distributed Systems
  10. Software Engineering: Fundamentals of Software Engineering, Software Requirements and Quality Assurance, Software Design, Estimation and Metrics, Software Testing, Software Maintenance and Configuration Management, Object Oriented Technology, HTML Basics, JavaScript-Basics, Java, XML
  11. Database Management System: 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, Data Warehousing & Mining, Enhanced Data Models, NoSQL
  12. Other Recurring CS Topics: Basics of Computer Graphics, 2-D Geometrical Transforms and Viewing, Multimedia
  13. Live Classes: ISRO Live classes
  14. Full Mock Test:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This educational video is a calculus lecture delivered by Yash Jain Sir, focusing on the advanced integration technique known as Integration by Parts. The session is designed to help students master the method of integrating products of functions by providing a structured approach to selecting the components u and v. The lecture begins with the derivation and presentation of the standard formula, followed by a detailed explanation of the "LIATE Rule," a mnemonic device used to determine the priority of functions. The instructor then reinforces these concepts by solving three distinct examples on the whiteboard, covering combinations of algebraic, trigonometric, logarithmic, and exponential functions. The video aims to clarify the often confusing process of choosing which function to differentiate and which to integrate.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video opens with a title card featuring the word "CALCULUS" amidst a background of complex mathematical equations like cos x = sum and i^2 = -1. The scene shifts to a classroom setting where Yash Jain Sir, identified by the text "YASH JAIN SIR KNOWLEDGE GATE EDUCATOR" on the screen, introduces the topic "Integration by Parts" written at the top of the whiteboard. He writes the integral formula int u * v dx = u int v dx - int u' (int v dx) dx in blue ink. He then simplifies this expression to the more compact form uv - int u'v dx. To address the common difficulty students face in choosing the correct functions, he writes the question "How to identify u & v?" in red ink. He introduces the "LIATE Rule" as the solution, writing it on the board with an arrow to indicate its importance in the upcoming lesson.

  2. 2:00 – 5:00 02:00-05:00

    The instructor uses a slide to explain the LIATE Rule in detail. The slide lists a specific hierarchy of functions: L for Logarithmic functions (e.g., ln(x), log_b(x)), I for Inverse Trigonometric functions (including hyperbolic analogues like arctan(x)), A for Algebraic functions (e.g., x^2, 3x^50), T for Trigonometric functions (including hyperbolic analogues like sin(x)), and E for Exponential functions (e.g., e^x, 19^x). He explains that this is a "rule of thumb" where the function appearing first in the list is chosen as u, and the function appearing last is chosen as v. He writes "LIATE" on the whiteboard and circles the 'A' and 'T' to emphasize their position. He notes that functions lower on the list generally have easier antiderivatives, which is the reasoning behind the rule. He also writes "integrate" under the 'E' to reinforce that the last function should be integrated. The slide text explicitly states: "The function which is to be v is whichever comes last in the list."

  3. 5:00 – 9:45 05:00-09:45

    The lecture moves to solving three specific integrals to demonstrate the rule. For the first example, int x sin x dx, he identifies x as u (Algebraic) and sin x as v (Trigonometric) based on the LIATE order. He applies the formula to get x int sin x - int 1 * int sin x. For the second example, int x^2 log x dx, he reverses the roles, choosing u = log x (Logarithmic) because it appears first in LIATE, and v = x^2 (Algebraic). He calculates the integral as log x * x^3/3 - int 1/x * x^3/3 dx. Finally, for int x e^x dx, he selects u = x (Algebraic) and v = e^x (Exponential), solving it to get x e^x - e^x + c. He factors the final result to e^x(x-1) + c. The board work shows the step-by-step application of the formula for each case.

The lesson effectively bridges the gap between theoretical formula memorization and practical application. By introducing the LIATE rule, the instructor provides a clear decision-making framework that simplifies the often confusing process of selecting u and v. The progression from the general formula to the specific rule, and finally to concrete examples, ensures that students understand not just how to apply the method, but why certain choices lead to simpler integrals. The use of visual aids and step-by-step board work reinforces the learning objectives, making the abstract concept of integration by parts more accessible and easier to apply in exam scenarios.

Loading lesson…