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
- Derivatives of Composite Functions
- Geometrical Meaning of Derivative
- Derivative of Inverse Function
- Logarithmic Differentiation
- Derivative of Implicit Functions
- Derivatives of Functions in Parametric Forms
- 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
- Basic Concepts of Differential Equations
- Order and Degree of a Differential Equation
- Formation of Differential Equations
- Methods of Solving Differential Equations> Homogeneous Differential Equations
- Methods of Solving Differential Equations>Linear Differential Equations
- Applications of Differential Equation
- Solution of a Differential Equation
- Overview of 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
Definition: Integration Using Partial Fraction
Integration by partial fractions is a method used to integrate rational functions, that is, functions of the form
Maharashtra State Board: Class 12
Types of Partial Fractions
| Type | Rational Form | Partial Form |
|---|---|---|
| Type I (Non-repeated linear factors) | \[\frac{\mathrm{p}x+\mathrm{q}}{(x-\mathrm{a})(x-\mathrm{b})}\] | \[\frac{\mathrm{A}}{x-\mathrm{a}}+\frac{\mathrm{B}}{x-\mathrm{b}}\] |
| \[\frac{\mathrm{p}x^{2}+\mathrm{q}x+\mathrm{r}}{(x-\mathrm{a})(x-\mathrm{b})(x-\mathrm{c})}\] | \[\frac{\mathrm{A}}{x-\mathrm{a}}+\frac{\mathrm{B}}{x-\mathrm{b}}+\frac{\mathrm{C}}{x-\mathrm{c}}\] | |
| Type II (Repeated linear factors) | \[\frac{\mathrm{p}x+\mathrm{q}}{\left(x-\mathrm{a}\right)^2}\] | \[\frac{\mathrm{A}}{(x-\mathrm{a})}+\frac{\mathrm{B}}{(x-\mathrm{a})^{2}}\] |
| \[\frac{\mathrm{p}x^{2}+\mathrm{q}x+\mathrm{r}}{\left(x-\mathrm{a}\right)^{2}\left(x-\mathrm{b}\right)}\] | \[\frac{\mathrm{A}}{(x-\mathrm{a})}+\frac{\mathrm{B}}{(x-\mathrm{a})^{2}}+\frac{\mathrm{C}}{(x-\mathrm{b})}\] | |
| Type III (Linear × Quadratic) | \[\frac{\mathrm{p}x^{2}+\mathrm{q}x+\mathrm{r}}{(x-\mathrm{a})(x^{2}+\mathrm{b}x+\mathrm{c})}\] | \[\frac{\mathrm{A}}{(x-\mathrm{a})}+\frac{\mathrm{B}x+\mathrm{C}}{(x^{2}+\mathrm{b}x+\mathrm{c})}\] |
Example 1
Find \[\int \frac{x^2 + 1}{x^2 - 5x + 6} dx\]
Solution:
Step 1: Divide
Here the integrand \[\frac{x^2 + 1}{x^2 - 5x + 6}\] is not proper rational function
Step 2: Factor denominator
\[x^2 + 1\] - \[x^2 - 5x + 6\] = 5x - 5
Let \[\frac{5x - 5}{(x - 2)(x - 3)} = \frac{\text{A}}{x - 2} + \frac{\text{B}}{x - 3}\]
Step 3: Find A and B
\[5x - 5 = \text{A}(x - 3) + \text{B}(x - 2)\]
Equating the coefficients of \[x\] and constant terms on both sides, we get \[\text{A} + \text{B} = 5\] and \[3\text{A} + 2\text{B} = 5\]. Solving these equations, we get
B = 5 − A
Substitute into the second equation:
3A + 2(5−A) = 5
3A + 10 − 2A = 5
A = −5 & B = 10
Thus, \[\frac{x^2 + 1}{x^2 - 5x + 6} = 1 - \frac{5}{x - 2} + \frac{10}{x - 3}\]
Step 4: Rewrite integral
Therefore, \[\int \frac{x^2 + 1}{x^2 - 5x + 6} dx = \int dx - 5 \int \frac{1}{x - 2} dx + 10 \int \frac{dx}{x - 3}\]
Step 5: Integrate
\[= x - 5 \log |x - 2| + 10 \log |x - 3| + \text{C}\].
Maharashtra State Board: Class 12
Key Points : Partial Fractions
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First check whether the rational function is proper or improper.
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Use long division before decomposition if the fraction is improper.
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Factorise the denominator completely before choosing partial fractions.
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For each distinct linear factor, use a constant numerator such as A, B, or C.
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For a repeated linear factor, include every power separately.
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For an irreducible quadratic factor, use a linear numerator of the form Bx + C.
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After decomposition, integrate each term separately using standard formulas.
