#### Topics

##### Relations and Functions

##### Relations and Functions

##### Inverse Trigonometric Functions

##### Algebra

##### Matrices

- Introduction of Operations on Matrices
- Inverse of a Matrix by Elementary Transformation
- Multiplication of Two Matrices
- Negative of Matrix
- Properties of Matrix Addition
- Transpose of a Matrix
- Subtraction of Matrices
- Addition of Matrices
- Symmetric and Skew Symmetric Matrices
- Types of Matrices
- Proof of the Uniqueness of Inverse
- Invertible Matrices
- Elementary Transformations
- Multiplication of Matrices
- Properties of Multiplication of Matrices
- Equality of Matrices
- Order of a Matrix
- Matrices Notation
- Introduction of Matrices
- Multiplication of a Matrix by a Scalar
- Properties of Scalar Multiplication of a Matrix
- Properties of Transpose of the Matrices

##### Calculus

##### Vectors and Three-dimensional Geometry

##### Determinants

- Applications of Determinants and Matrices
- Elementary Transformations
- Inverse of a Square Matrix by the Adjoint Method
- Properties of Determinants
- Determinant of a Square Matrix
- Determinants of Matrix of Order One and Two
- Introduction of Determinant
- Area of a Triangle
- Minors and Co-factors
- Determinant of a Matrix of Order 3 × 3
- Rule A=KB

##### Linear Programming

##### Continuity and Differentiability

- Derivative - Exponential and Log
- Concept of Differentiability
- Proof Derivative X^n Sin Cos Tan
- Infinite Series
- Higher Order Derivative
- Algebra of Continuous Functions
- Continuous Function of Point
- Mean Value Theorem
- Second Order Derivative
- Derivatives of Functions in Parametric Forms
- Logarithmic Differentiation
- Exponential and Logarithmic Functions
- Derivatives of Implicit Functions
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Composite Functions - Chain Rule
- Concept of Continuity

##### Probability

##### Applications of Derivatives

- Maximum and Minimum Values of a Function in a Closed Interval
- Maxima and Minima
- Simple Problems on Applications of Derivatives
- Graph of Maxima and Minima
- Approximations
- Tangents and Normals
- Increasing and Decreasing Functions
- Rate of Change of Bodies or Quantities
- Introduction to Applications of Derivatives

##### Sets

- Sets

##### Integrals

- Definite Integrals Problems
- Indefinite Integral Problems
- Comparison Between Differentiation and Integration
- Geometrical Interpretation of Indefinite Integrals
- Integrals of Some Particular Functions
- Indefinite Integral by Inspection
- Some Properties of Indefinite Integral
- Integration Using Trigonometric Identities
- Introduction of Integrals
- Evaluation of Definite Integrals by Substitution
- Properties of Definite Integrals
- Fundamental Theorem of Calculus
- Definite Integral as the Limit of a Sum
- Evaluation of Simple Integrals of the Following Types and Problems
- Methods of Integration: Integration by Parts
- Methods of Integration: Integration Using Partial Fractions
- Methods of Integration: Integration by Substitution
- Integration as an Inverse Process of Differentiation

##### Applications of the Integrals

##### Differential Equations

- Linear Differential Equations
- Solutions of Linear Differential Equation
- Homogeneous Differential Equations
- Differential Equations with Variables Separable Method
- Formation of a Differential Equation Whose General Solution is Given
- General and Particular Solutions of a Differential Equation
- Order and Degree of a Differential Equation
- Basic Concepts of Differential Equation
- Procedure to Form a Differential Equation that Will Represent a Given Family of Curves

##### Vectors

- Direction Cosines
- Properties of Vector Addition
- Geometrical Interpretation of Scalar
- Scalar Triple Product of Vectors
- Vector (Or Cross) Product of Two Vectors
- Scalar (Or Dot) Product of Two Vectors
- Position Vector of a Point Dividing a Line Segment in a Given Ratio
- Multiplication of a Vector by a Scalar
- Addition of Vectors
- Introduction of Vector
- Magnitude and Direction of a Vector
- Basic Concepts of Vector Algebra
- Vectors and Their Types
- Components of Vector
- Section Formula
- Vector Joining Two Points
- Vectors Examples and Solutions
- Projection of a Vector on a Line
- Introduction of Product of Two Vectors

##### Three - Dimensional Geometry

- Three - Dimensional Geometry Examples and Solutions
- Introduction of Three Dimensional Geometry
- Equation of a Plane Passing Through Three Non Collinear Points
- Relation Between Direction Ratio and Direction Cosines
- Intercept Form of the Equation of a Plane
- Coplanarity of Two Lines
- Distance of a Point from a Plane
- Angle Between Line and a Plane
- Angle Between Two Planes
- Angle Between Two Lines
- Vector and Cartesian Equation of a Plane
- Shortest Distance Between Two Lines
- Equation of a Line in Space
- Direction Cosines and Direction Ratios of a Line
- Equation of a Plane in Normal Form
- Equation of a Plane Perpendicular to a Given Vector and Passing Through a Given Point
- Plane Passing Through the Intersection of Two Given Planes

##### Linear Programming

##### Probability

- Variance of a Random Variable
- Probability Examples and Solutions
- Conditional Probability
- Multiplication Theorem on Probability
- Independent Events
- Bayes’ Theorem
- Random Variables and Its Probability Distributions
- Mean of a Random Variable
- Bernoulli Trials and Binomial Distribution
- Introduction of Probability
- Properties of Conditional Probability

## Text

The explain the some properties of indefinite integrals. **I)** The process of differentiation and integration are inverses of each other in the sense of the following results :

`d/(dx) int` f(x)dx = f(x)

and `int f '(x) dx = f(x) + C` , where C is any arbitrary constant.

**Proof:** Let F be any anti derivative of f, i.e.,

`d/(dx)` F(x) = f(x)

Then `int` f(x)dx = F(x) +C

Therefore `d/(dx) int` f(x) dx = `d/(dx)`(f(x)+C)

= `d/(dx)` F(x)=f(x)

Similarly, we note that

f'(x) = `d/(dx)` f(x)

and hence `int`f'(x) dx = f(x) +C

where C is arbitrary constant called constant of integration.

**II)** Two indefinite integrals with the same derivative lead to the same family of curves and so they are equivalent. **Proof:** Let f and g be two functions such that

`d/(dx) int f(x)dx = d/(dx) int g(x)dx`

or `d/dx [int f(x) dx - int g(x) dx]` = 0

Hence ∫f(x) dx - ∫g(x) dx = C, where C is any real number

or `int f(x)dx = int g(x)dx + C `

So the families of curves `{int f(x) dx + C_1, C_1 ∈ R}`

and `{int g(x) dx + C_2 , C_2 ∈ R}` are identical.

Hence, in this sense, `int f(x) dx ` and `int g(x) dx` are equivalent.

**III)** ∫[f(x) + g(x)]dx = ∫f(x)dx + ∫ g(x) dx**Proof:** By Property (I), we have

`d/(dx)int[f(x) + g(x)dx] = f(x) +g(x)` ...(1)

On the otherhand, we find that

`d/(dx)[ int f(x) dx + int g(x) dx] = d/(dx) int f(x) dx + d/(dx) int g(x) dx`

=f(x) + g(x) ...(2)

Thus, in view of Property (II), it follows by (1) and (2) that

`int (f(x) + g(x))dx = int f(x) dx + int g(x) dx .`

**IV)** For any real number k, `int k f(x) dx = k int f(x) dx`**Proof:** By the Property (I),

`d/(dx) int k f(x) dx = k f(x).`

Also `d/(dx) [k int f(x)dx] = k d/(dx) int f(x) dx` = k f(x)

Therefore, using the Property (II), we have `int k f(x) dx = k int f(x) dx .`

**V)** Properties (III) and (IV) can be generalised to a finite number of functions `f_1, f_2, ..., f_n` and the real numbers, `k_1, k_2, ..., k_n` giving

`int [k_1f_1(x) + k_2f_2(x) + ...+k_nf_n (x)] dx`

= `k_1 int f_1(x) dx +k_2 int f_2 (x) dx + ... + k_n int f_n (x) dx`