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Integration

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Fundamental Law of Calculus

Put very simply:

int{,~a,~b,}fract{d ~f,d~x}( ~x ) _ d~x _ = _ _ ~f ( ~b ) - ~f ( ~a )

Note that _ ~f ( ~b ) - ~f ( ~a ) _ is often written _ [ _ ~f ( ~x ) _ ]__~a^~b

So we can often treat integration as the reverse of differentiation. This gives us the following simple integrals:

Simple Integrals

The limits ~a and ~b are left out for the sake of clarity. Written in this form (but usually without the square brackets), these are known as indefinite integrals.

~{∫} ~c d~x _ = _ [ ~c~x ]

~{∫} ~c Df( ~x )d~x _ = _ [ ~c f( ~x ) ]

~{∫} ~x^{~m} d~x _ = _ [ ~x^{~m + 1} ./ (~m + 1) ] , _ _ ~m != -1

~{∫} ~e ^~x d~x _ = _ [ ~e ^~x ]

~{∫} ~x^{-1} d~x _ = _ [ ln ~x ]

~{∫} cos( ~x ) d~x _ = _ [ sin( ~x ) ]

~{∫} sin( ~x ) d~x _ = _ [ - cos( ~x ) ]

~{∫} sec^2( ~x ) d~x _ = _ [ tan( ~x ) ]

~{∫} 1 ./ &sqrt.${~a&powtwo. - ~x&powtwo.} d~x _ = _ [ sin^{-1}( ~x/~a ) ]

~{∫} - 1 ./ &sqrt.${~a&powtwo. - ~x&powtwo.} d~x _ = _ [ cos^{-1}( ~x/~a ) ]

~{∫} 1 ./ ( ~a&powtwo. - ~x&powtwo. ) d~x _ = _ [ tan^{-1}( ~x/~a ) ]

Indefinite Integrals

Differentiation is a mapping from a set of functions to another. (The first set must be continuous and differentiable etc.). This mapping is well defined, in the sense that each (differentiable) function has a derivative. However it is not one-to-one, as a derivative can be a derivative for a family of functions which differ only by a constant:

fract{d ~f( ~x ) + ~c,d~x} _ = _ fract{d ~f( ~x ),d~x} , _ _ any constant _ ~c , _ _ since _ fract{d ~c,d~x} _ = _ 0

The integral sign without the limits is used to indicate this reverse mapping, and these are known as #~{indefinite integrals}. To make it clear that the result is not unique, the arbitrary constant _ ~c _ is included, so we write, for example

int{ sin( ~x ) ,,,d~x} _ = _ - cos( ~x ) + ~c

In this context ~c is called the #~{constant of integration}. _ Note that when integrating between limits the constant of integration disappears

int{,~a,~b,}fract{d ~f,d~x}( ~x ) _ d~x _ = _ script{sqrb{ _ ~f ( ~x ) + ~c _ },,,~b,~a} _ = _ _ ~f ( ~b ) + ~c - ~f ( ~a ) - ~c _ = _ _ ~f ( ~b ) - ~f ( ~a )

Integration as Linear Operator

Integration is a linear operator, i.e:

~{∫} ~aF( ~x ) + ~bG( ~x ) d~x _ = _ ~a ~{∫} F( ~x ) d~x + ~b ~{∫} G( ~x ) d~x