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Example:Integration by Parts

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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




Integration by parts


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 '(xdx and dv = g'(xdx, then integration by parts states that


or simply:

(Our solved example in mathguru.com uses this concept)



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 ′(xdx and dv = g′(xdx, 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.




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.