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Integration
Introduction:
"To Integrate" in calculus has two meanings. One will be clearly understood in finding
areas bounded by curves, volumes of various solids, lengths of curves etc.
The second meaning of "integrate" is to find a function whose derivative is given.
In the beginning we will discuss the second aspect of integration.
In algebra the operations of addition and subtraction, multiplication and division
are reverse operations of each other. In the same manner, integration or primitive
is the reverse process of differentiation.
Definition
Let F(x) be any differentiable function of x. If there is a function f(x) such that
Then F(x) is called the anti-derivative or a primitive or simply an integral of f(x) with respect to x.
Symbolically, this is written as ∫ f(x) dx = F (x) and read as "Integration of f(x) with respect to x is F(x)."
If F(x) is an antiderivative of f(x), then F(x) + C is also an antiderivative where C is any constant.
This is because:
Note that this antiderivative F(x) is unique i.e any function y whose derivative is f(x) can differ from F only by a constant.
Hence the formula F(x) + C gives all the antiderivatives of f(x) is called the indefinite integral of f(x) with respect to x and this is written as:
Here the function f(x) is called the integrand of the integral and C is called the constant of integration. Also dx tells us that the integration is with respect to x.
|
d
dx |
[ F (x) ] | = | f (x) |
Then F(x) is called the anti-derivative or a primitive or simply an integral of f(x) with respect to x.
Symbolically, this is written as ∫ f(x) dx = F (x) and read as "Integration of f(x) with respect to x is F(x)."
If F(x) is an antiderivative of f(x), then F(x) + C is also an antiderivative where C is any constant.
This is because:
|
d
dx |
[ F (x) + C ] | = |
dF
dx |
+ |
dC
dx |
= | f (x) + 0 | = | f (x) |
Note that this antiderivative F(x) is unique i.e any function y whose derivative is f(x) can differ from F only by a constant.
Hence the formula F(x) + C gives all the antiderivatives of f(x) is called the indefinite integral of f(x) with respect to x and this is written as:
∫ f(x) dx = F (x) + C
Here the function f(x) is called the integrand of the integral and C is called the constant of integration. Also dx tells us that the integration is with respect to x.
Examples:
|
d
dx |
(x2) | = | 2x | => | ∫ 2x dx | = | x2+ C |
|
d
dx |
(sin x) | = | cos x | => | ∫ cos x dx | = | sin x + C |
Hence from the known formulae of derivatives we will list the formulae for standard
integrals.
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