Vipul at 02:42, 27 August 2011

2011-08-27T02:42:02Z

← Older revision		Revision as of 02:42, 27 August 2011
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	This function, denoted <math>\tan^3</math>, is defined as the [[defining ingredient::composite of two functions\|composite]] of the [[defining ingredient::cube function]] and the [[defining ingredient::tangent function]].		This function, denoted <math>\tan^3</math>, is defined as the [[defining ingredient::composite of two functions\|composite]] of the [[defining ingredient::cube function]] and the [[defining ingredient::tangent function]].

			==Differentiation==

			===First derivative===

			To calculate the first derivative of <math>\tan^3</math>, we note that the function is the [[composite of two functions\|composite]] of the [[cube function]] and the [[tangent function]], and differentiate using the [[chain rule for differentiation]]:

			<math>\frac{d}{dx}((\tan x)^3) = (3(\tan x)^2)\frac{d}{dx}(\tan x) = 3\tan^2x \sec^2x</math>

			===Higher derivatives===

			{{fillin}}

			==Integration==

			===First antiderivative===

			We use the identity:

			<math>\! \tan^2x = \sec^2x - 1</math>

			as follows:

			<math>\int \tan^3x \, dx = \int \tan x \tan^2x \, dx = \int \tan x (\sec^2x - 1) \, dx = \int \tan x \sec^2x \, dx - \int \tan x \, dx</math>

			The term <math>\tan x \sec^2x</math> integrates to <math>(\tan^2x)/2</math> (we can see this, for instance, by putting <math>u = \tan x</math>, see also [[tan.sec^2#Integration]]). The term <math>\tan x</math> integrates to <math>-\ln\|\cos x\|</math> (See [[tan#Integration]]). We thus get:

			<math>\frac{\tan^2x}{2} + \ln\|\cos x\| + C</math>

			Note that the value of <math>C</math> is constant within every ''connected component'' of the domain, but could differ between different connected components of the domain.

			===Higher antiderivatives===

			It is not possible to integrate this function repeatedly in terms of [[elementary function]]s, because <math>\ln\|\cos x\|</math> cannot be integrated in terms of elementary functions.

Vipul: Created page with "{{particular function}} {{angular function radian convention}} ==Definition== This function, denoted $\tan^3$ , is defined as the [[defining ingredient::composite of..."

2011-08-26T17:47:55Z

Created page with "{{particular function}} {{angular function radian convention}} ==Definition== This function, denoted <math>\tan^3</math>, is defined as the [[defining ingredient::composite of..."

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{{particular function}}

{{angular function radian convention}}

==Definition==

This function, denoted <math>\tan^3</math>, is defined as the [[defining ingredient::composite of two functions|composite]] of the [[defining ingredient::cube function]] and the [[defining ingredient::tangent function]].

Tangent-cubed function - Revision history

Vipul at 02:42, 27 August 2011

Vipul: Created page with "{{particular function}} {{angular function radian convention}} ==Definition== This function, denoted \tan^3, is defined as the [[defining ingredient::composite of..."

Vipul: Created page with "{{particular function}} {{angular function radian convention}} ==Definition== This function, denoted $\tan^3$ , is defined as the [[defining ingredient::composite of..."