Topics
Mathematical Logic
- Concept of Statements
- Truth Value of Statement
- Logical Connective, Simple and Compound Statements
- Statement Patterns and Logical Equivalence
- Tautology, Contradiction, and Contingency
- Duality
- Quantifier and Quantified Statements in Logic
- Negations of Compound Statements
- Converse, Inverse, and Contrapositive
- Algebra of Statements
- Application of Logic to Switching Circuits
- Overview of Mathematical Logic
Matrices
- Elementry Transformations
- Properties of Matrix Multiplication
- Application of Matrices
- Applications of Determinants and Matrices
- Overview of Matrices
Trigonometric Functions
- Trigonometric Equations and Their Solutions
- Solutions of Triangle
- Inverse Trigonometric Functions
- Overview of Trigonometric Functions
Pair of Straight Lines
- Combined Equation of a Pair Lines
- Homogeneous Equation of Degree Two
- Angle between lines represented by ax2 + 2hxy + by2 = 0
- General Second Degree Equation in x and y
- Equation of a Line in Space
- Overview of Pair of Straight Lines
Vectors
Line and Plane
- Vector and Cartesian Equations of a Line
- Distance of a Point from a Line
- Distance Between Skew Lines and Parallel Lines
- Equation of a Plane
- Angle Between Planes
- Coplanarity of Two Lines
- Distance of a Point from a Plane
- Overview of Line and Plane
Linear Programming
Differentiation
- Differentiation
- Derivatives of Composite Functions - Chain Rule
- Geometrical Meaning of Derivative
- Derivatives of Inverse Functions
- Logarithmic Differentiation
- Derivatives of Implicit Functions
- Derivatives of Parametric Functions
- Higher Order Derivatives
- Overview of Differentiation
Applications of Derivatives
- Applications of Derivatives in Geometry
- Derivatives as a Rate Measure
- Approximations
- Rolle's Theorem
- Lagrange's Mean Value Theorem (LMVT)
- Increasing and Decreasing Functions
- Maxima and Minima
- Overview of Applications of Derivatives
Indefinite Integration
Definite Integration
- Definite Integral as Limit of Sum
- Integral Calculus
- Methods of Evaluation and Properties of Definite Integral
- Overview of Definite Integration
Application of Definite Integration
- Application of Definite Integration
- Area Bounded by the Curve, Axis and Line
- Area Between Two Curves
- Overview of Application of Definite Integration
Differential Equations
- Differential Equations
- Order and Degree of a Differential Equation
- Formation of Differential Equations
- Homogeneous Differential Equations
- Linear Differential Equations
- Application of Differential Equations
- Solution of a Differential Equation
- Overview of Differential Equations
Probability Distributions
- Random Variables and Its Probability Distributions
- Types of Random Variables
- Probability Distribution of Discrete Random Variables
- Probability Distribution of a Continuous Random Variable
- Variance of a Random Variable
- Expected Value and Variance of a Random Variable
- Overview of Probability Distributions
Binomial Distribution
- Bernoulli Trial
- Binomial Distribution
- Mean of Binomial Distribution (P.M.F.)
- Variance of Binomial Distribution (P.M.F.)
- Bernoulli Trials and Binomial Distribution
- Overview of Binomial Distribution
Estimated time: 15 minutes
Maharashtra State Board: Class 12
Formula: Elementary Integration Formulae
| Function | Integral |
|---|---|
| \[x^{n}\] | \[\frac{x^{n+1}}{n+1}+C\] |
| $$\int (ax + b)^n dx$$ | $$\int (ax + b)^n dx$$\[\frac{x^{n+1}}{n+1}+C\] |
| \[\int a^xdx\] | \[\frac{a^x}{\log a}+C\] |
| \[\int A^{ax+b}dx\] | \[\frac{A^{ax+b}}{a\log A}+C\] |
| \[\int e^xdx\] | \[e^{x}+C\] |
| \[\int e^{ax+b}dx\] | \[\frac{e^{ax+b}}{a}+C\] |
| \[\int\cos xdx\] | \[\sin x+C\] |
| \[\int\cos(ax+b)dx\] | \[\frac{\sin(ax+b)}{a}+C\] |
| $$\int \sin x dx$$ | \[-\cos x+C\] |
| \[\int\sin(ax+b)dx\] | \[-\frac{\cos(ax+b)}{a}+C\] |
| \[\int\sec^2xdx\] | \[\tan x+C\] |
| \[\int\sec^2(ax+b)dx\] | \[\frac{\tan(ax+b)}{a}+C\] |
| $$\int \sec x \tan x dx$$ | \[\sec x+C\] |
| \[\int\sec(ax+b)\tan(ax+b)dx\] | \[\frac{\sec(ax+b)}{a}+C\] |
| \[\int\operatorname{cosec}x\cot xdx\] | \[-\operatorname{cosec}x+C\] |
| \[\int\operatorname{cosec}(ax+b)\cot(ax+b)dx\] | \[-\frac{\mathrm{cosec}(ax+b)}{a}+C\] |
| \[\int\mathrm{cosec}^2xdx\] | \[-\cot x+C\] |
| \[\int\mathrm{cosec}^2(ax+b)dx\] | \[-\frac{\cot(ax+b)}{a}+C\] |
| \[\int\frac{1}{x}dx\] | \[\log x+C\] |
| \[\int\frac{1}{ax+b}dx\] | \[\frac{1}{a}\log(ax+b)+C\] |
Maharashtra State Board: Class 12
Formula: Integration by Substitution
If x = φ(t) is a differentiable function of t, then \[\int f\left(x\right)\quad dx=\int f\left[\phi\left(t\right)\right]\phi^{\prime}\left(t\right)dt.\]
| \[\int f\left(ax+b\right)\quad dx=g\left(ax+b\right)\frac{1}{a}+c\] |
| \[\int[f(x)]^n\cdot f^{\prime}(x)dx=\frac{[f(x)]^{n+1}}{n+1}+c\] |
| \[\int\frac{f^{\prime}(x)}{f(x)}dx=\log|f(x)|+C\] |
Maharashtra State Board: Class 12
Formula: Some Special Integrals
| No. | Standard Integral | Result |
|---|---|---|
| 1 | \[\int\frac{1}{x^2+a^2}dx\] | \[\frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right)+C\] |
| 2 | \[\int\frac{1}{x^2-a^2}dx\] | \[\frac{1}{2a}\log\left(\frac{x-a}{x+a}\right)+c\] |
| 3 | \[\int\frac{1}{a^2-x^2}\quad dx\] | \[\frac{1}{2a}\log\left(\frac{a+x}{a-x}\right)+C\] |
| 4 | \[\int\frac{1}{\sqrt{a^2-x^2}}\quad dx\] | \[\sin^{-1}\left(\frac{X}{a}\right)+c\] |
| 5 | \[\int\frac{1}{\sqrt{x^2-a^2}}dx\] | \[\log\left(x+\sqrt{x^{2}-a^{2}}\right)+c\] |
| 6 | \[\int\frac{1}{\sqrt{a^2+x^2}}\quad dx\] | \[\log\left(x+\sqrt{a^{2}+x^{2}}\right)+c\] |
| 7 | \[\int\frac{1}{x\sqrt{x^2-a^2}}dx\] | \[\frac{1}{a}\sec^{-1}\left(\frac{x}{a}\right)+c\] |
Maharashtra State Board: Class 12
Formula: Standard Substitutions
| Form in integral | Best substitution |
|---|---|
| \[\sqrt{a^2-x^2}\] | (or a cosθ) |
| \[\sqrt{a^2+x^2}\] | x = a tanθ |
| \[\sqrt{x^2-a^2}\] | x = a secθ |
| \[\sqrt{\frac{a-x}{a+x}}\] | x = a cos2θ |
Maharashtra State Board: Class 12
Formula: Integration by Parts
\[\int u\mathrm{~}dv=uv-\int v\mathrm{~}du\]
Maharashtra State Board: Class 12
Key Points: LIATE Rule
| Order | Function Type |
|---|---|
| L | Logarithmic |
| I | Inverse Trigonometric |
| A | Algebraic |
| T | Trigonometric |
| E | Exponential |
Maharashtra State Board: Class 12
Formula: Integration by Partial Fractions
| Rational Form | Partial Fraction Form |
|---|---|
| \[\frac{P(x)}{(x-a)(x-b)(x-c)}\] | \[\frac{A}{x-a}+\frac{B}{x-b}+\frac{C}{x-c}\] |
| \[\frac{P(x)}{(x-a)^2(x-b)}\] | \[\frac{A}{x-a}+\frac{B}{(x-a)^2}+\frac{C}{x-b}\] |
| \[\frac{P(x)}{(x-a)(x^2+bx+c)}\] | \[\frac{A}{x-a}+\frac{Bx+C}{x^2+bx+c}\] |
| \[\frac{P(x)}{(x^2+bx+c)^2}\] | \[\frac{Ax+B}{x^2+bx+c}+\frac{Cx+D}{(x^2+bx+c)^2}\] |
