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Example:Evaluate Integral using Substitution

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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 [ab] 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 [ab], then, once an antiderivative F of ƒ is known, the definite integral of ƒ over that interval is given by




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)





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


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.




The above explanation is copied from Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.