7.9 Fundamental Theorem of Calculus > No. 19

Find the definite integration of the given function.

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Explanation:

Integral

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

$Description: \int_a^b \! f(x)\,dx \,$

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:

$Description: F = \int f(x)\,dx.$

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

$Description: \int_a^b \! f(x)\,dx = F(b) - F(a)\,$

http://en.wikipedia.org/wiki/Integral

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 $Description: I\subseteq {\mathbb{R}}$ be an interval and $Description: g : [a,b] \to I$ be a continuously differentiable function. Suppose that $Description: f : I \to \mathbb{R}$ is a continuous function. Then

$Description: \int_a^b f(g(t))g'(t)\, dt = \int_{g(a)}^{g(b)} f(x)\,dx.$

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 $Description: g'\,$ 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

$Description: \int_{g(a)}^{g(b)} f(x)\,dx$

and

$Description: \int_a^b f(g(t))g'(t)\,dt$

in fact exist, and it remains to show that they are equal.

Since ƒ is continuous, it possesses an antiderivative F. The composite function $Description: F\circ g$ is then defined. Since F and g are differentiable, the chain rule gives

$Description: (F \circ g)'(t) = F'(g(t))g'(t) = f(g(t))g'(t).$

Applying the fundamental theorem of calculus twice gives

$Description: \begin{align} \int_a^b f(g(t))g'(t)\,dt & {} = (F \circ g)(b) - (F \circ g)(a) \\ & {} = F(g(b)) - F(g(a)) \\ & {} = \int_{g(a)}^{g(b)} f(x)\,dx, \end{align}$

which is the substitution rule.

http://en.wikipedia.org/wiki/Integration_by_substitution

Fundamental theorem of calculus

The fundamental theorem of calculus specifies the relationship between the two central operations of calculus: differentiation and integration.

The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integrationcan be reversed by a differentiation. The first part is also important because it guarantees the existence of antiderivatives for continuous functions.

The second part, sometimes called the second fundamental theorem of calculus, allows one to compute the definite integral of a function by using any one of its infinitely many antiderivatives. This part of the theorem has invaluable practical applications, because it markedly simplifies the computation of definite integrals.

Formal statements

There are two parts to the Fundamental Theorem of Calculus. Loosely put, the first part deals with the derivative of an antiderivative, while the second part deals with the relationship between antiderivatives and definite integrals.

First part

This part is sometimes referred to as the First Fundamental Theorem of Calculus.

Let ƒ be a continuous real-valued function defined on a closed interval [a, b]. Let F be the function defined, for all x in [a, b], by

$Description: F(x) = \int_a^x f(t)\, dt\,.$

Then, F is continuous on [a, b], differentiable on the open interval (a, b), and

$Description: F'(x) = f(x)\,$

for all x in (a, b).

Corollary

The fundamental theorem is often employed to compute the definite integral of a function ƒ for which an antiderivative g is known. Specifically, if ƒ is a real-valued continuous function on [a, b], and g is an antiderivative of ƒ in [a, b], then

$Description: \int_a^b f(x)\, dx = g(b)-g(a).$

The corollary assumes continuity on the whole interval. This result is strengthened slightly in the following theorem.

Second part

This part is sometimes referred to as the Second Fundamental Theorem of Calculus or the Newton-Leibniz Axiom.

Let ƒ be a real-valued function defined on a closed interval [a, b] that admits an antiderivative g on [a, b]. That is, ƒ and g are functions such that for all x in [a, b],

$Description: f(x) = g'(x).\$

If ƒ is integrable on [a, b] then

$Description: \int_a^b f(x)\,dx\, = g(b) - g(a).$

Notice that the Second part is somewhat stronger than the Corollary because it does not assume that ƒ is continuous.

Note that when an antiderivative g exists, then there are infinitely many antiderivatives for ƒ, obtained by adding to g an arbitrary constant. Also, by the first part of the theorem, antiderivatives of ƒ always exist when ƒ is continuous.

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

http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus

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.