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Heather's workspace

Integration by parts 1

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Author

Heather Driscoll

Project

Heather's workspace

Licence

Ability levels

England schools
- Primary school
- Key Stage 1
- Key Stage 2
- Key Stage 3
- Key Stage 4 / GCSE
- A-Level
England university
- Transition to university
- Stage 1
- Stage 2
- Stage 3
- Postgraduate
Scotland schools
- 1st Level
- 2nd Level
- 3rd/4th Level
- National 4 & 5
- Higher
- Advanced Higher

Taxonomy: mathcentre

mathcentre's taxonomy.

1 - Numbers and Computation

1.1 - Number Concepts

1.1.1 - Natural

1.1.2 - Integers

1.1.3 - Rational

1.1.4 - Irrational

1.1.5 - Algebraic

1.1.6 - Real

1.1.7 - Complex

1.1.8 - Famous Numbers

1.1.8.1 - 0

1.1.8.2 - pi

1.1.8.3 - e

1.1.8.4 - i

1.1.8.5 - Golden Mean

1.1.9 - Symbols, constants

1.2 - Arithmetic

1.2.1 - Operations

1.2.1.1 - Addition

1.2.1.2 - Subtraction

1.2.1.3 - Multiplication

1.2.1.4 - Division

1.2.1.5 - Roots

1.2.1.6 - Factorials

1.2.1.7 - Factoring

1.2.1.8 - Properties of Operations

1.2.1.9 - Estimation

1.2.2 - Fractions

1.2.2.1 - Addition

1.2.2.2 - Subtraction

1.2.2.3 - Multiplication

1.2.2.4 - Division

1.2.2.5 - Ratio and Proportion

1.2.2.6 - Equivalent Fractions

1.2.3 - Decimals

1.2.3.1 - Addition

1.2.3.2 - Subtraction

1.2.3.3 - Multiplication

1.2.3.4 - Division

1.2.3.5 - Percents

1.2.4 - Comparison of numbers

1.2.5 - Exponents

1.2.5.1 - Multiplication

1.2.5.2 - Division

1.2.5.3 - Powers

1.2.5.4 - Integer Exponents

1.2.5.5 - Rational Exponents

1.2.6 - Logarithms

1.2.6.1 - Addition

1.2.6.2 - Subtraction

1.2.6.3 - Multiplication

1.2.6.4 - Natural logarithms

1.2.6.5 - Change of base

1.3 - Patterns and Sequences

1.3.1 - Number Patterns

1.3.2 - Fibonacci Sequence

1.3.3 - Arithmetic Sequence

1.3.4 - Geometric Sequence

1.4 - Measurement

1.4.1 - Units of Measurement

1.4.1.1 - Metric System

1.4.1.2 - Standard Units

1.4.1.3 - Nonstandard Units

1.4.2 - Linear Measure

1.4.2.1 - Distance

1.4.2.2 - Circumference

1.4.2.3 - Perimeter

1.4.3 - Area

1.4.3.1 - Area of Polygons

1.4.3.2 - Area of Circles

1.4.3.3 - Surface Area

1.4.3.4 - Nonstandard Shapes

1.4.4 - Volume

1.4.5 - Weight and Mass

1.4.6 - Temperature

1.4.7 - Time

1.4.8 - Speed

1.4.9 - Money

1.4.10 - Scale

2 - Logic and Foundations

2.1 - Logic

2.1.1 - Logic

2.1.2 - Propositional and Predicate Logic

2.1.3 - Methods of Proof

2.2 - Set Theory

2.2.1 - Sets and Set Operations

2.2.2 - Relations and Functions

2.2.3 - Cardinality

2.2.4 - Axiom of Choice

2.3 - Computability and Decidability

2.4 - Model Theory

3 - Algebra and Number Theory

3.1 - Algebra

3.1.1 - Graphing Techniques

3.1.2 - Algebraic Manipulation

3.1.2.1 - Collecting like terms

3.1.2.2 - Rules of indices for single term expressions

3.1.2.2.1 - Surds

3.1.2.3 - Polynomial expressions

3.1.2.3.1 - Multiplying out brackets

3.1.2.3.1.1 - Multiplying a single-term expression and bracket

3.1.2.3.1.2 - Multiplying two brackets

3.1.2.3.1.3 - Pure powers

3.1.2.3.1.4 - The difference of two squares

3.1.2.3.1.5 - Multiple brackets

3.1.2.3.2 - Factoring polynomials

3.1.2.3.2.1 - Single term common factor

3.1.2.3.2.2 - Quadratic expressions

3.1.2.3.2.3 - Higher powers

3.1.2.3.2.4 - Multiple variable expressions

3.1.2.4 - Algebraic fractions

3.1.2.4.1 - Cancelling

3.1.2.4.1.1 - Cancelling fractions involving factorising

3.1.2.4.2 - Multiplying by -1

3.1.2.4.3 - Adding and subtracting algebraic fractions

3.1.2.4.4 - Multiplying and dividing algebraic fractions

3.1.2.4.5 - Partial fractions

3.1.2.4.6 - Rationalizing surd expressions

3.1.2.5 - Completing the square of a quadratic expression

3.1.2.6 - Rearranging equations

3.1.2.6.1 - Substituting from one formula into another

3.1.2.6.2 - Making a specified letter the subject of a formula

3.1.2.6.3 - Substitution

3.1.2.7 - De-nesting surds

3.1.3 - Functions

3.1.3.1 - Linear

3.1.3.2 - Quadratic

3.1.3.3 - Polynomial

3.1.3.4 - Rational

3.1.3.5 - Exponential

3.1.3.6 - Logarithmic

3.1.3.7 - Piece-wise

3.1.3.8 - Step

3.1.3.9 - Hyperbolics

3.1.3.9.1 - Hyperbolic functions

3.1.3.9.2 - Inverse hyperbolic functions

3.1.3.9.3 - Hyperbolic identities

3.1.4 - Equations

3.1.4.1 - Linear

3.1.4.2 - Quadratic

3.1.4.3 - Polynomial

3.1.4.4 - Rational

3.1.4.5 - Exponential

3.1.4.6 - Logarithmic

3.1.4.7 - Systems

3.1.5 - Inequalities

3.1.6 - Matrices

3.1.7 - Sequences and Series

3.1.8 - Algebraic Proof

3.2 - Linear Algebra

3.2.1 - Systems of Linear Equations

3.2.2 - Matrix algebra

3.2.3 - Vectors in R3

3.2.4 - Vector Spaces

3.2.5 - Linear Transformations

3.2.6 - Eigenvalues and Eigenvectors

3.2.7 - Inner Product Spaces

3.3 - Abstract Algebra

3.3.1 - Groups

3.3.2 - Rings and Ideals

3.3.3 - Fields

3.3.4 - Galois Theory

3.3.5 - Multilinear Algebra

3.4 - Number Theory

3.4.1 - Integers

3.4.2 - Primes

3.4.2.1 - Divisibility

3.4.2.2 - Factorization

3.4.2.3 - Distributions of Primes

3.4.3 - Congruences

3.4.4 - Diophantine Equations

3.4.5 - Irrational Numbers

3.4.6 - Famous Problems

3.4.7 - Coding Theory

3.4.8 - Cryptography

3.5 - Category Theory

3.6 - K-Theory

3.7 - Homological Algebra

3.8 - Modular Arithmetic

4 - Discrete Mathematics

4.1 - Cellular Automata

4.2 - Combinatorics

4.2.1 - Combinations

4.2.2 - Permutations

4.3 - Game Theory

4.4 - Algorithms

4.5 - Recursion

4.6 - Graph Theory

4.7 - Linear Programming

4.8 - Order and Lattices

4.9 - Theory of Computation

4.10 - Chaos

5 - Geometry and Topology

5.1 - Geometric Proof

5.2 - Plane Geometry

5.2.1 - Measurement

5.2.2 - Lines and Planes

5.2.3 - Angles

5.2.4 - Triangles

5.2.4.1 - Properties

5.2.4.2 - Congruence

5.2.4.3 - Similarity

5.2.4.4 - Pythagorean Theorem

5.2.5 - Polygons

5.2.5.1 - Properties

5.2.5.2 - Regular

5.2.5.3 - Irregular

5.2.5.4 - Congruence

5.2.5.5 - Similarity

5.2.6 - Circles

5.2.7 - Patterns

5.2.7.1 - Geometric Patterns

5.2.7.2 - Tilings and Tessellations

5.2.7.3 - Symmetry

5.2.7.4 - Golden Ratio

5.2.8 - Transformations

5.2.8.1 - Translation

5.2.8.2 - Rotation

5.2.8.3 - Reflection

5.2.8.4 - Scaling

5.3 - Solid Geometry

5.3.1 - Dihedral Angles

5.3.2 - Spheres

5.3.3 - Cones

5.3.4 - Cylinders

5.3.5 - Pyramids

5.3.6 - Prisms

5.3.7 - Polyhedra

5.4 - Analytic Geometry

5.4.1 - Cartesian Coordinates

5.4.2 - Lines

5.4.3 - Circles

5.4.4 - Planes

5.4.5 - Conics

5.4.6 - Polar Coordinates

5.4.7 - Parametric Curves

5.4.8 - Surfaces

5.4.9 - Distance Formula

5.5 - Projective Geometry

5.6 - Differential Geometry

5.7 - Algebraic Geometry

5.8 - Topology

5.8.1 - Point Set Topology

5.8.2 - General Topology

5.8.3 - Differential Topology

5.8.4 - Algebraic Topology

5.9 - Trigonometry

5.9.1 - Angles

5.9.2 - Trigonometric Functions

5.9.3 - Inverse Trigonometric Functions

5.9.4 - Trigonometric Identities

5.9.5 - Trigonometric Equations

5.9.6 - Roots of Unity

5.9.7 - Spherical Trigonometry

5.10 - Fractal Geometry

6 - Calculus

6.1 - Single Variable

6.1.1 - Functions

6.1.2 - Limits

6.1.3 - Continuity

6.1.4 - Differentiation

6.1.5 - Integration

6.1.6 - Series

6.2 - Several Variables

6.2.1 - Functions of Several Variables

6.2.2 - Limits

6.2.3 - Continuity

6.2.4 - Partial Derivatives

6.2.5 - Multiple integrals

6.2.6 - Taylor Series

6.3 - Advanced Calculus

6.3.1 - Vector Valued Functions

6.3.2 - Line Integrals

6.3.3 - Surface Integrals

6.3.4 - Stokes Theorem

6.3.5 - Curvilinear Coordinates

6.3.6 - Linear spaces

6.3.7 - Fourier Series

6.3.8 - Orthogonal Functions

6.4 - Tensor Calculus

6.5 - Calculus of Variations

6.6 - Operational Calculus

7 - Analysis

7.1 - Real Analysis

7.1.1 - Metric Spaces

7.1.2 - Convergence

7.1.3 - Continuity

7.1.4 - Differentiation

7.1.5 - Integration

7.1.6 - Measure Theory

7.2 - Complex Analysis

7.2.1 - Convergence

7.2.2 - Infinite Series

7.2.3 - Analytic Functions

7.2.4 - Integration

7.2.5 - Contour Integrals

7.2.6 - Conformal Mappings

7.2.7 - Several Complex Variables

7.3 - Numerical Analysis

7.3.1 - Computer Arithmetic

7.3.2 - Solutions of Equations

7.3.3 - Solutions of Systems

7.3.4 - Interpolation

7.3.5 - Numerical Differentiation

7.3.6 - Numerical Integration

7.3.7 - Numerical Solutions of ODEs

7.3.8 - Numerical Solutions of PDEs

7.4 - Integral Transformations

7.4.1 - Fourier Transforms

7.4.2 - Laplace Transforms

7.4.3 - Hankel Transforms

7.4.4 - Wavelets

7.4.5 - Other Transforms

7.5 - Signal Analysis

7.5.1 - Sampling Theory

7.5.2 - Filters

7.5.3 - Noise

7.5.4 - Data Compression

7.5.5 - Image Processing

7.6 - Functional Analysis

7.6.1 - Hilbert Spaces

7.6.2 - Banach Spaces

7.6.3 - Topological Spaces

7.6.4 - Locally Convex Spaces

7.6.5 - Bounded Operators

7.6.6 - Spectral Theorem

7.6.7 - Unbounded Operators

7.7 - Harmonic Analysis

7.8 - Global Analysis

8 - Differential and Difference Equations

8.1 - Ordinary Differential Equations

8.1.1 - First Order

8.1.2 - Second Order

8.1.3 - Linear Oscillations

8.1.4 - Nonlinear Oscillations

8.1.5 - Systems of Differential Equations

8.1.6 - Sturm Liouville Problems

8.1.7 - Special Functions

8.1.8 - Power Series Methods

8.1.9 - Laplace Transforms

8.2 - Partial Differential Equations

8.2.1 - First Order

8.2.2 - Elliptic

8.2.3 - Parabolic

8.2.4 - Hyperbolic

8.2.5 - Integral Transforms

8.2.6 - Integral Equations

8.2.7 - Potential Theory

8.2.8 - Nonlinear Equations

8.2.9 - Symmetries and Integrability

8.3 - Difference Equations

8.3.1 - First Order

8.3.2 - Second Order

8.3.3 - Linear Systems

8.3.4 - Z Transforms

8.3.5 - Orthogonal Polynomials

8.4 - Dynamical Systems

8.4.1 - 1D Maps

8.4.2 - 2D Maps

8.4.3 - Lyapunov Exponents

8.4.4 - Bifurcations

8.4.5 - Fractals

8.4.6 - Differentiable Dynamics

8.4.7 - Conservative Dynamics

8.4.8 - Chaos

8.4.9 - Complex Dynamical Systems

9 - Statistics and Probability

9.1. - Data Collection

9.1.1. - Experimental Design

9.1.2. - Sampling and Surveys

9.1.3. - Data and Measurement Issues

9.2. - Data Summary and Presentation

9.2.1. - Summary Statistics

9.2.1.1. - Measures of Central Tendency

9.2.1.2. - Measures of Spread

9.2.2. - Data Representation

9.2.2.1. - Graphs and Plots

9.2.2.2. - Tables

9.3. - Statistical Inference and Techniques

9.3.1. - Sampling Distributions

9.3.2. - Regression and Correlation

9.3.3. - Confidence Intervals

9.3.4. - Hypothesis Tests

9.3.5. - Statistical Quality Control

9.3.6. - Non-parametric Techniques

9.3.7. - Multivariate Techniques

9.3.8. - Survival Analysis

9.3.9. - Bayesian Statistics

9.4. - Probability

9.4.3 - Limit Theorems

9.4.3.1 - Central Limit Theorem

9.4.3.2 - Law of Large Numbers

9.4.5 - Stochastic Processes

9.4.5.1 - Brownian Motion

9.4.5.2 - Markov Chains

9.4.5.3 - Queuing Theory

9.4.6 - Probability Measures

9.4.7 - Simulation

9.4.1. - Elementary Probability

9.4.1.1. - Sample Space and Events

9.4.1.2. - General Rules

9.4.1.3. - Combinations and Permutations

9.4.1.4. - Random Variables

9.4.2. - Univariate Distributions

9.4.2.1. - Discrete Distributions

9.4.2.2. - Continuous Distributions

9.4.2.3. - Expected Value

9.4.4. - Multivariate Distributions

9.4.4.1 - Joint

9.4.4.2 - Conditional

9.4.4.3 - Expectations

10.0 - Applied Mathematics

10.1 - Mathematical Physics

10.2 - Mathematical Economics

10.3 - Mathematical Biology

10.4 - Mathematics for Business

10.5 - Engineering Mathematics

10.5.1 - Mathematics for Materials

10.6 - Mathematical Sociology

10.7 - Mathematics for Social Sciences

10.7.1 - Mathematics for Social Policy and Social Work

10.8 - Mathematics for Computer Science

10.9 - Mathematics for Humanities

10.9.1 - Mathematics for Art, Design and Communication

10.9.2 - Mathematics for Built Environment

10.9.3 - Mathematics for History, Classics and Archaeology

10.9.4 - Mathematics for Law

10.9.5 - Mathematics for Performing Arts

10.9.6 - Mathematics for Education

10.9.7 - Mathematics for Languages, Linguistics and Area Studies

10.9.7.1 - Mathematics for English

10.9.8 - Mathematics for Philosophical and Religious Studies

10.10 - Consumer Mathematics

10.11 - Mathematics for Economics

10.12 - Mathematics for Life Sciences

10.12.1 - Mathematics for Geography, Earth and Environmental Sciences

10.12.2 - Mathematics for Bioscience

10.12.3 - Mathematics for Health Sciences and Practice

10.12.4 - Mathematics for Medicine, Dentistry and Veterinary Medicine

10.13 - Mathematics for Psychology

10.14 - Mathematics for Physical Sciences

10.15 - Mathematics for Sociology, Anthropology and Politics

10.16 - Mathematics for Hospitality, Leisure, Sport and Tourism

11.0 - Mathematics History

11.1 - General

11.2 - Famous Problems

11.3 - Biographies of Mathematicians

12.0 - Mathematical Education

Taxonomy: Kind of activity

1 - Create

1.1 - Give an example

1.2 - Give a starting point

1.3 - Give an argument

2 - Evaluate

2.1 - Make a decision

2.2 - Classify an object

3 - Analyze

3.1 - Interpret a situation

3.2 - Select from a list of options

3.3 - Identify an error

4 - Apply

4.1 - Use formulas

4.2 - Perform a routine calculation or algorithm

4.3 - Apply an algorithm

5 - Understand

5.1 - Use an appropriate formula

5.2 - Identify a component

6 - Remember

6.1 - Recall a fact

Taxonomy: Context

1 - Mathematics

1.1 - Pure

1.2 - Applied

1.3 - For...

1.3.1 - Accountancy

1.3.2 - Business

1.3.3 - Psychology

1.3.4 - Science

1.3.4.1 - Biomedicine

1.3.4.2 - Chemistry

1.3.4.3 - Biology

2 - Statistics

2.1 - Bayesian

2.2 - Frequentist

3 - Physics

3.1 - Theoretical

4 - Engineering

4.1 - Civil

4.2 - Electrical

4.3 - Chemical

5 - Chemistry

Contributors

Heather Driscoll

History

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Checkpoint description

Describe what's changed since the last checkpoint.

○

Heather Driscoll 3 years, 8 months ago

Published this.

○

Heather Driscoll 3 years, 8 months ago

Created this.

Name	Status	Author	Last Modified
Integration by parts 1	draft	Heather Driscoll	03/09/2021 14:53
Integration by parts 2	draft	Heather Driscoll	03/09/2021 15:00

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Mathematical expression

Mathematical expression

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Solve the following integral using the 'by parts' method:

$\int x\sin{(2x)}\, dx$

Enter your answer in the same way you would in MATLAB.

Fractions can be entered using the \ symbol.

Variables must be multiplied using the * symbol, for example $xy$ should be entered as x*y

Functions must include brackets, for example $\sin{x}$ should be entered as sin(x)

Don't forget to include the "+ C" as it is an indefinite integral.

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Correct answer

$\frac{ 1 }{ 4 } \sin \left ( 2 x \right ) - \frac{ 1 }{ 2 } x \cos \left ( 2 x \right ) + C$

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Example,

$\displaystyle \int 2xe^x\,dx$
$u = 2x \rightarrow \frac{du}{dx}=2$
$\frac{dv}{dx}=e^x \rightarrow v = \int e^x.dx = e^x$
$\int 2xe^x.dx = (2x)\left(e^x\right) - \int \left(e^x \right) \left(2\right).dx$
$=2xe^x - 2\int e^x.dx$
$=2xe^x - 2e^x + C$
$=2e^x(x-1) + C$

The tabular method could also be used. In this approach, one function is differentiated until it reaches zero. The other is integrated the corresponding number of times. An alternative sign approach is then used to multiple down the table and form the answer. Please see the notes from AMR129 for an example of this.

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This question is used in the following exam:

AMR229 Week 1 - review by Heather Driscoll in Heather's workspace.

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