Integration is an important concept in mathematics and, together with differentiation, is one of the
two main operations in calculus. Given a function ƒ of a real variable x and an interval [a, b] of
the real line, the definite integral
is defined informally to be the net signed area of
the region in the xy-plane
bounded by the graph of ƒ,
the x-axis, and the
vertical lines x = a and x = b.
The term integral may also refer to the notion of antiderivative, a function F whose derivative is the given function ƒ. In this case, it is called
an indefinite integral and
is written:
The integrals
discussed in this article are termed definite integrals. Some authors maintain a
distinction between antiderivatives and indefinite integrals.
The principles
of integration were formulated independently by Isaac Newton and Gottfried Leibniz in the late 17th century.
Through the fundamental theorem of calculus, which they independently
developed, integration is connected with differentiation: if ƒ is a continuous real-valued
function defined on a closed interval [a, b], then, once an antiderivative F of ƒ is known, the definite integral
of ƒ over that interval is given by
http://en.wikipedia.org/wiki/Integral
In calculus, and
more generally in mathematical
analysis, integration by parts is a rule that transforms the integral of products of functions into other
(ideally simpler) integrals. The rule can be derived in one line by simply
integrating the product rule of differentiation.
If u = f(x), v = g(x),
and the differentials du = f '(x) dx and dv = g'(x) dx,
then integration by parts states that
or simply:
(Our solved example in mathguru.com uses this concept)
Rule
Suppose f(x)
and g(x)
are two continuously differentiable functions.
The product rule states
Integrating both sides gives
Rearranging terms
From the above one can derive the integration by parts rule, which
states that, given an interval with endpoints a and b,
with the common notation
The rule is
shown to be true by using the product rule for derivatives and the fundamental theorem of
calculus. Thus
In the
traditional calculus curriculum, the rule is often stated using indefinite integrals in the form
or, if u = f(x), v = g(x)
and the differentials du = f ′(x) dx and dv = g′(x) dx,
then it is in the form most often seen:
This formula can be interpreted to mean that the area under the
graph of a function u(v)
is the same as the area of the rectangle u v minus the area above the graph.
The original integral contains the derivative of g; in order to be able to apply
the rule, the antiderivative g must be found, and then the resulting
integral ∫g f ′ dx must be evaluated.
One can also formulate a discrete analogue for sequences, called summation by parts.
http://en.wikipedia.org/wiki/Integration_by_parts
The above explanation is copied from
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