Topics
Relations and Functions
Relations and Functions
Algebra
Inverse Trigonometric Functions
- Basics of Inverse Trigonometric Functions
- Domain, Range & Principal Value
- Graphs of Inverse Trigonometric Functions
- Properties of Inverse Trigonometric Functions
- Overview of Inverse Trigonometric Functions
Calculus
Matrices
Vectors and Three-dimensional Geometry
Determinants
- Determinant of a Matrix
- Expansion of Determinant
- Area of Triangle using Determinant
- Minors and Co-factors
- Adjoint & Inverse of Matrix
- Applications of Determinants and Matrices
- Overview of Determinants
Continuity and Differentiability
- Continuous and Discontinuous Functions
- Algebra of Continuous Functions
- Concept of Differentiability
- Derivative of Composite Functions
- Derivative of Implicit Functions
- Derivative of Inverse Function
- Exponential and Logarithmic Functions
- Logarithmic Differentiation
- Derivatives of Functions in Parametric Forms
- Second Order Derivative
- Overview of Continuity and Differentiability
Linear Programming
Probability
Applications of Derivatives
Sets
Integrals
- Introduction of Integrals
- Integration as an Inverse Process of Differentiation
- Properties of Indefinite Integral
- Methods of Integration> Integration by Substitution
- Methods of Integration>Integration Using Trigonometric Identities
- Methods of Integration> Integration Using Partial Fraction
- Methods of Integration> Integration by Parts
- Integrals of Some Particular Functions
- Definite Integrals
- Fundamental Theorem of Integral Calculus
- Evaluation of Definite Integrals by Substitution
- Properties of Definite Integrals
- Overview of Integrals
Applications of the Integrals
Differential Equations
- Basic Concepts of Differential Equations
- Order and Degree of a Differential Equation
- General and Particular Solutions of a Differential Equation
- Methods of Solving Differential Equations> Variable Separable Differential Equations
- Methods of Solving Differential Equations> Homogeneous Differential Equations
- Methods of Solving Differential Equations>Linear Differential Equations
- Overview of Differential Equations
Vectors
- Basic Concepts of Vector Algebra
- Direction Ratios, Direction Cosine & Direction Angles
- Types of Vectors in Algebra
- Algebra of Vector Addition
- Multiplication in Vector Algebra
- Components of Vector in Algebra
- Vector Joining Two Points in Algebra
- Section Formula in Vector Algebra
- Product of Two Vectors
- Overview of Vectors
Three - Dimensional Geometry
Linear Programming
Probability
Key Points: Properties of Definite Integrals
(i)\[\int_{a}^{b}\mathrm{f}\left(x\right)\mathrm{d}x=\int_{a}^{b}\mathrm{f}\left(t\right)\mathrm{d}t\]
(ii)\[\int_{\mathrm{a}}^{\mathrm{b}}\mathrm{f}\left(x\right)\mathrm{d}x=-\int_{\mathrm{b}}^{\mathrm{a}}\mathrm{f}\left(x\right)\mathrm{d}x\]
(iii)\[\int_{a}^{b}f\left(x\right)\mathrm{d}x=\int_{a}^{c}f\left(x\right)\mathrm{d}x+\int_{c}^{b}f\left(x\right)\mathrm{d}x,a<c<b\]
(iv)\[\int_{0}^{a}\mathrm{f}\left(x\right)\mathrm{d}x=\int_{0}^{a}\mathrm{f}\left(\mathrm{a}-x\right)\mathrm{d}x\]
(v)\[\int_{\mathrm{a}}^{\mathrm{b}}\mathrm{f}\left(x\right)\mathrm{d}x=\int_{\mathrm{a}}^{\mathrm{b}}\mathrm{f}\left(\mathrm{a}+\mathrm{b}-x\right)\mathrm{d}x\]
(vi)\[\int_{0}^{2a}\mathrm{f}\left(x\right)\mathrm{d}x=\int_{0}^{a}\left[\mathrm{f}\left(x\right)+\mathrm{f}\left(2\mathrm{a}-x\right)\right]\mathrm{d}x\]
(vii)\[\int_{0}^{2a}\mathrm{f}\left(x\right)\mathrm{d}x=2\int_{0}^{a}\mathrm{f}\left(x\right)\mathrm{d}x,\] if f(2a − x) = f(x)
= 0, if f(2a − x) = −f(x)
(viii)$$\int_{-a}^{a} f(x) dx = \begin{cases} 2 \int_{0}^{a} f(x) dx, & \text{if } f(x) \text{ is an even function} \\ & \text{i.e., } f(-x) = f(x) \\ 0, & \text{if } f(x) \text{ is an odd function} \\ & \text{i.e., } f(-x) = -f(x) \end{cases}$$
