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
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.
General Formula:
If \[x = g(t), \ dx = g'(t) dt\] then \[\int f(x) dx = \int f(g(t))g'(t) dt\]
Maharashtra State Board: Class 12
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 |
Stepwise Method
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Identify the complicated part of the integrand.
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Choose a substitution such as \(u = g(x)\).
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Differentiate to get \(du\) and rewrite \(dx\) accordingly.
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Express the entire integral in terms of the new variable.
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Integrate in the simpler form.
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Replace the new variable by the original expression.
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Add the constant of integration for indefinite integrals.
Example 1
Find the following integrals:
- \[\int \sin^3 x \cos^2 x dx\]
- \[\int \frac{\sin x}{\sin (x + a)} dx\]
Solution:
(i) We have
\[\int \sin^3 x \cos^2 x dx = \int \sin^2 x \cos^2 x (\sin x) dx\]
\[= \int (1 - \cos^2 x) \cos^2 x (\sin x) dx\]
Put \[t = \cos x\] so that \[dt = -\sin x dx\]
Therefore, \[\int \sin^2 x \cos^2 x (\sin x) dx = -\int (1 - t^2) t^2 dt\]
\[= -\int (t^2 - t^4) dt = -\left( \frac{t^3}{3} - \frac{t^5}{5} \right) + \text{C}\]
\[= -\frac{1}{3} \cos^3 x + \frac{1}{5} \cos^5 x + \text{C}\]
(ii) Put \[x + a = t\]. Then \[dx = dt\].
Therefore
\[\int \frac{\sin x}{\sin (x + a)} dx = \int \frac{\sin (t - a)}{\sin t} dt\]
\[= \int \frac{\sin t \cos a - \cos t \sin a}{\sin t} dta \]
\[= \cos a \int dt - \sin a \int \cot t dta\]
\[= (\cos a) t - (\sin a) [\log |\sin t| + \text{C}_1]\]
\[= (\cos a) (x + a) - (\sin a) [\log |\sin (x + a)| + \text{C}_1][= x \cos a + a \cos a - (\sin a) \log |\sin (x + a)| - \text{C}_1 \sin a\]
Hence, a \[\int \frac{\sin x}{\sin (x + a)} dx = x \cos a - \sin a \log |\sin (x + a)| + \text{C}\],
where, \[\text{C} = -\text{C}_1 \sin a + a \cos a\], is another arbitrary constant.
Maharashtra State Board: Class 12
Key Points: Standard Substitution
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Integration by substitution is the reverse process of the chain rule.
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Choose the substitution so that the integral becomes simpler, not more complicated.
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Always rewrite both the function and \(dx\) in terms of the new variable.
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For indefinite integrals, back-substitute and add \(C\).
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For definite integrals, limits should also be changed if the solution is continued in the new variable.
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Trigonometric substitution is mainly used for radicals involving \(a^2-x^2\), \(x^2+a^2\), and \(x^2-a^2\).
