Topics
Mathematical Logic
- Statements and Truth Values in Mathematical Logic
- Logical Connectives
- Tautology, Contradiction, and Contingency
- Quantifier, Quantified and Duality 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
Trigonometric Functions
Pair of Straight Lines
Vectors
Line and Plane
Linear Programming
Differentiation
- Introduction & Derivatives of Some Standard Functions
- Derivative of Composite Functions
- Geometrical Meaning of Derivative
- Derivative of Inverse Function
- 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 Two Curves
- Overview of Application of Definite Integration
Differential Equations
Probability Distributions
- 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
Estimated time: 6 minutes
Maharashtra State Board: Class 12
Definition: Integration by Substitution
Integration by substitution is a method in which we replace a part of the integral by a new variable to simplify the integration.
Maharashtra State Board: Class 12
Key Points: Standard Substitution
| Sr. No. | Integrand Form | Substitution |
|---|---|---|
| i | \[\sqrt{\mathrm{a}^2-x^2},\frac{1}{\sqrt{\mathrm{a}^2-x^2}},\mathrm{a}^2-x^2\] | x = a sinθ or x = a cosθ |
| ii | \[\sqrt{x^2+\mathrm{a}^2},\frac{1}{\sqrt{x^2+\mathrm{a}^2}},x^2+\mathrm{a}^2\] | x = a tanθ |
| iii | \[\sqrt{x^{2}-a^{2}},\frac{1}{\sqrt{x^{2}-a^{2}}},x^{2}-a^{2}\] | x = a secθ |
| iv | \[\sqrt{\frac{x}{a+x}},\sqrt{\frac{a+x}{x}},\]\[\sqrt{x(a+x)},\frac{1}{\sqrt{x(a+x)}}\] | x = a tan²θ |
| v | \[\sqrt{\frac{x}{a-x}},\sqrt{\frac{a-x}{x}},\]\[\sqrt{x(a-x)},\frac{1}{\sqrt{x(a-x)}}\] | x = a sin²θ |
| vi | \[\sqrt{\frac{x}{x-a}},\sqrt{\frac{x-a}{x}},\]\[\sqrt{x(x-\mathrm{a})},\frac{1}{\sqrt{x(x-\mathrm{a})}}\] | x = a sec²θ |
| vii | \[\sqrt{\frac{\mathrm{a}-x}{\mathrm{a}+x}},\sqrt{\frac{\mathrm{a}+x}{\mathrm{a}-x}}\] | x = a cos 2θ |
| viii | \[\sqrt{\frac{x-\alpha}{\beta-x}},\sqrt{(x-\alpha)(\beta-x)},\]\[(\beta>\alpha)\] | x = α cos²θ + β sin²θ |
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