Topics
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Relations and Functions
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Inverse Trigonometric Functions
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Calculus
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Linear Programming
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Applications of Derivatives
Integrals
- Introduction of Integrals
- Integration as an Inverse Process of Differentiation
- Properties of Indefinite Integral
- Methods of Integration> Integration by Substitution
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Three - Dimensional Geometry
Linear Programming
Probability
Estimated time: 8 minutes
CBSE: Class 12
Introduction
Integration is a fundamental concept in calculus used to find a function when its derivative is known. It is called the inverse process of differentiation because it reverses the operation of finding derivatives.
In simple words, if differentiation tells how a quantity changes, integration helps recover the original quantity from that rate of change. This idea forms the base of many later topics in mathematics, physics, economics, and engineering.
CBSE: Class 12
Definition: Antiderivative
If the derivative of a function F(x) is f(x), then F(x) is called an antiderivative or integral of f(x). The set of all such antiderivatives is written as:
\[\int f(x) dx = F(x) + C\]
where C is an arbitrary constant called the constant of integration.
CBSE: Class 12
Formula: Integral Formulas
| No. | Derivatives | Integrals (Anti-derivatives) |
|---|---|---|
| (i) | \[\frac{d}{dx} \left( \frac{x^{n+1}}{n+1} \right) = x^n\]; | \[\int x^n dx = \frac{x^{n+1}}{n+1} + \text{C}, n \neq -1\] |
| \[\frac{d}{dx} (x) = 1\]; | \[\int dx = x + \text{C}\] | |
| (ii) | \[\frac{d}{dx} (\sin x) = \cos x\]; | \[\int \cos x dx = \sin x + \text{C}\] |
| (iii) | \[\frac{d}{dx} (-\cos x) = \sin x\]; | \[\int \sin x dx = -\cos x + \text{C}\] |
| (iv) | \[\frac{d}{dx} (\tan x) = \sec^2 x\]; | \[\int \sec^2 x dx = \tan x + \text{C}\] |
| (v) | \[\frac{d}{dx} (-\cot x) = \text{cosec}^2 x\]; | \[\int \text{cosec}^2 x dx = -\cot x + \text{C}\] |
| (vi) | \[\frac{d}{dx} (\sec x) = \sec x \tan x\]; | \[\int \sec x \tan x dx = \sec x + \text{C}\] |
| (vii) | \[\frac{d}{dx} (-\text{cosec} x) = \text{cosec} x \cot x\]; | \[\int \text{cosec} x \cot x dx = -\text{cosec} x + \text{C}\] |
| (viii) | \[\frac{d}{dx} (\sin^{-1} x) = \frac{1}{\sqrt{1-x^2}}\]; | \[\int \frac{dx}{\sqrt{1-x^2}} = \sin^{-1} x + \text{C}\] |
| (ix) | \[\frac{d}{dx} (-\cos^{-1} x) = \frac{1}{\sqrt{1-x^2}}\]; | \[\int \frac{dx}{\sqrt{1-x^2}} = -\cos^{-1} x + \text{C}\] |
| (x) | \[\frac{d}{dx} (\tan^{-1} x) = \frac{1}{1+x^2}\]; | \[\int \frac{dx}{1+x^2} = \tan^{-1} x + \text{C}\] |
| (xi) | \[\frac{d}{dx} (e^x) = e^x\]; | \[\int e^x dx = e^x + \text{C}\] |
| (xii) | \[\frac{d}{dx}\left(\log|x|\right)=\frac{1}{x};\] | \[\int\frac{1}{x}dx=\log|x|+\mathrm{C}\] |
| (xiii) | \[\frac{d}{dx} \left( \frac{a^x}{\log a} \right) = a^x\]; | \[\int a^x dx = \frac{a^x}{\log a} + \text{C}\] |
CBSE: Class 12
Key Points: Integration as an Inverse Process of Differentiation
-
Integration is the inverse process of differentiation.
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The result of indefinite integration is called the antiderivative or primitive.
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General form: ∫f(x) dx = F(x) + C.
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The constant CC must always be added in indefinite integrals.
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