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
Integrals of simple functions
C is used for an arbitrary constant of integration that can only be determined if something about
the value of the integral at some point is known. Thus each function has an
infinite number of antiderivatives.
Rational functions
These rational functions have a
non-integrable singularity at 0 for a ≤ −1.
(Cavalieri's quadrature formula)
(Our solved example in mathguru.com uses this concept)
http://en.wikipedia.org/wiki/Lists_of_integrals
Integration by substitution
In calculus, integration by substitution is a method for finding antiderivatives and integrals. Using the fundamental theorem of calculus often requires finding an
antiderivative. For this and other reasons, integration by substitution is an
important tool for mathematicians. It is the counterpart to the chain rule of differentiation.
Let
be an interval and be a continuously differentiable function. Suppose that is
a continuous function. Then
Relation
to the fundamental theorem of calculus
Integration by substitution can be derived from the fundamental theorem of calculus as follows. Let ƒ and g be two functions satisfying the above
hypothesis that ƒ is continuous on I and is continuous on the closed interval [a,b].
Then the function f(g(t))g'(t) is also continuous on [a,b]. Hence the integrals
and
in fact exist, and it remains to show that
they are equal.
Since ƒ is
continuous, it possesses an antiderivative F. The composite
function is then
defined. Since F and g are differentiable, the chain
rule gives
Applying the fundamental
theorem of calculus twice gives
which is the substitution
rule.
http://en.wikipedia.org/wiki/Integration_by_substitution
The
above explanation is copied from Wikipedia, the free encyclopedia and is
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License.