Integration by Substitution – Transform Methods :: Danielitte

Convex Quadrilateral

Segment of Circle

Sector of Circle

Regular Poligon of N Sides

Sperical Segment

Sperical Sector

Frustum of Right Circular Cone

Triangular Prism

Arithmetic Progression

Geometric Progression

Decimal Logarithm

Natural Logarithm

Euler's Formula

Trigonometric Functions For A Right Triangle

Trigonometric Table

Multiple Angle Formulas

Powers Of Trigonometric Functions

Formulas With T=tan(x/2)

Addition Formulas

Sum Of Trigonometric Functions

Product Of Trigonometric Functions

Half Angle Formulas

Angles Of A Plane Triangle

Sides And Angles Of A Plane Triangle

Relationships Among Trigonometric Fuctions

Linear Equation

System of Two Linear Equation

Quadratic Equation

Exponential Equation

Logarithmic Equation

Trigonometric Equation Cos

Trigonometric Equation Sin

Trigonometric Equation Tan

Trigonometric Equation Cotan

Linear Inequation

Quadratic Inequation

Exponential Inequation

Logarithmic Inequation

Trigonometric Inequation Cos

Trigonometric Inequation Sin

Trigonometric Inequation Tan

Trigonometric Inequation Cotan

Horizontal Shifting

Vertical Shifting

Equation of Line

Equation of Circle

Equation of Line Joining Two Points A,B

Equation of Sphere Center at M and Radius R in Rectangular Coordinates

Equation of Ellipsoid With Center M and Semi-Axes A,B,C

Elliptic Cylinder With Axis as Z Axis

Elliptic Cone With Axis as Z Axis

Hyperboloid of One Sheets

Hyperboloid of Two Sheets

Elliptic Paraboloid

Hyperbolic Paraboloid

Differentiation

Indefinite Integrals

Integrals By Partial Functions

Integrals Involving Roots

Integrals Involving Trigonometric Functions

Transformations

Definite Integrals

Transpose Matrix

Addition And Substraction Of Matrices

Multiplication Of Matrices

Determinant Of Matrices

Inverse Of Matrix

Equation In Matrix Form

Properties Of Matrix Calculations

Relative Complement of A in B

Absolute Complement

Symmetric Difference

Operations On Sets

Standard Deviation

Root Mean Square

Normal Distribution(gaussion Distribution)

Exponential Distribution

Poisson Distribution

Uniforn Distribution

Real Form Of Fourier Series

Parseval's Theorem

Fourier Transform

Fourier Symmetry Relationships

Fourier Transform Pairs

Substitution(Frequency Shifting)

Translation(Time Shifting)

Laplace Transform Pairs

Integration - Transformations

Transformations

Integration by Substitution Techniques

These transformations help simplify complex integrals by changing variables. Substitution is a powerful method for evaluating integrals involving algebraic, trigonometric, exponential, or logarithmic expressions.

\[ \int F(ax + b) \, dx = \frac{1}{a} \int F(u) \, du, \quad u = ax + b \]
\[ \int F(\sqrt{ax + b}) \, dx = \frac{2}{a} \int u F(u) \, du, \quad u = \sqrt{ax + b} \]
\[ \int F(\sqrt[n]{ax + b}) \, dx = \frac{n}{a} \int u^{n-1} F(u) \, du, \quad u = \sqrt[n]{ax + b} \]
\[ \int F(\sqrt{a^2 - x^2}) \, dx = a \int F(a \cos u) \cos u \, du, \quad x = a \sin u \]
\[ \int F(\sqrt{a^2 + x^2}) \, dx = a \int F(a \sec u) \sec^2 u \, du, \quad x = a \tan u \]
\[ \int F(\sqrt{x^2 - a^2}) \, dx = a \int F(a \tan u) \sec u \tan u \, du, \quad x = a \sec u \]
\[ \int F(e^{ax}) \, dx = \frac{1}{a} \int \frac{F(u)}{u} \, du, \quad u = e^{ax} \]
\[ \int F(\ln x) \, dx = \int F(u) e^u \, du, \quad u = \ln x \]

Terminology

Substitution: A method to simplify integration by changing variables.
Radical Transform: Used when integrand involves square roots or higher roots.
Trigonometric Substitution: Replacing expressions like \( \sqrt{a^2 - x^2} \) with trigonometric identities to ease integration.
Exponential & Log Substitution: Applied when integrand involves \( e^{ax} \) or \( \ln x \).

Applications

Solving integrals in calculus where direct integration is not possible.
Used in physics for changing coordinate systems (e.g., polar to Cartesian).
Modeling exponential growth or decay in biology, chemistry, and economics.
Solving engineering problems involving trigonometric curves or transformations.