Mathematics Revision Notes of Definite Integrals and Its Applications For NDA

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Revision Notes of Definite Integrals and Its Applications For NDA

Definite Integral

Let `f(x)` be the primitive or anti-derivative of a function `g(x)` define on `[a, b]`, i.e. `d/(dx) f(x) = g(x)`, then definite integral of `g(x)` over ` [a, b]` is denoted by `int _a^b g(x) dx` and defined as `[f(b)- f(a)]`.

`:. int_a^b g(x) dx = f(b) -f(a)`

Here, `a` and `b` are called the limits of integration, where `a` is called the lower limit and `b` is called the upper limit.

NOTE `=>` In definite integral, there is no need to keep the constant of integration.
`=> int_a^b f(x)dx` is read as the integral from `a` to `b`.

Let `f(x)` be the primitive or anti-derivative of a function `g(x)` define on `[a, b]`, i.e. `d/(dx) f(x) = g(x)`, then definite integral of `g(x)` over ` [a, b]` is denoted by `int _a^b g(x) dx` and defined as `[f(b)- f(a)]`.

`:. int_a^b g(x) dx = f(b) -f(a)`

Here, `a` and `b` are called the limits of integration, where `a` is called the lower limit and `b` is called the upper limit.

NOTE `=>` In definite integral, there is no need to keep the constant of integration.
`=> int_a^b f(x)dx` is read as the integral from `a` to `b`.

Properties of Definite Integral

`(A)text(Properties )`:

P-1: `int_a^b f(x) dx =int_a^b f(t) dt`;

P-2: `int_a^bf(x) dx =-int_b^a f(x) d(x)`

P-3: `int_a^b f(x) = int_a^c f(x) +int_c^b f(x) dx` provided `f` has a piece wise continuity or when `f` is not uniformly defined in `(a, b)`

Integral is broken at points of discontinuity or at the points where definition of `f` changes.

P-4: `int_-a^a f(x) dx =int _0^a (f(x) +f(-x))dx = [tt( (0,text(if) f(x)quad is quad odd),( 2 int_0^a f(x) dx, text(if) f(x)quad isquad even) )`

P-5: `int_a^b f(x) dx = int_a^b f(a+b-x) dx ` or `int_0^a f(x) dx =int_0^a f (a-x)dx`

P-6: `int_0^(2a) f(x) dx =int_0^a f(x) dx+int_0^a f(2a-x) dx => [tt( (0,text(if) f(2a-x) =-f(x)), (2 int_0^a f(x) dx , text(if) f (2a-x) =f(x)))`

P-7 : `int_0^(nT)f(x) dx =n int_0^Tf(x) dx` where `f(T+x)=f(x) n in I`

`(A)text(Properties )`:

P-1: `int_a^b f(x) dx =int_a^b f(t) dt`;

P-2: `int_a^bf(x) dx =-int_b^a f(x) d(x)`

P-3: `int_a^b f(x) = int_a^c f(x) +int_c^b f(x) dx` provided `f` has a piece wise continuity or when `f` is not uniformly defined in `(a, b)`

Integral is broken at points of discontinuity or at the points where definition of `f` changes.

P-4: `int_-a^a f(x) dx =int _0^a (f(x) +f(-x))dx = [tt( (0,text(if) f(x)quad is quad odd),( 2 int_0^a f(x) dx, text(if) f(x)quad isquad even) )`

P-5: `int_a^b f(x) dx = int_a^b f(a+b-x) dx ` or `int_0^a f(x) dx =int_0^a f (a-x)dx`

P-6: `int_0^(2a) f(x) dx =int_0^a f(x) dx+int_0^a f(2a-x) dx => [tt( (0,text(if) f(2a-x) =-f(x)), (2 int_0^a f(x) dx , text(if) f (2a-x) =f(x)))`

P-7 : `int_0^(nT)f(x) dx =n int_0^Tf(x) dx` where `f(T+x)=f(x) n in I`

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