Tangent-squared function - Revision history

Vipul: /* Integration */

2012-02-13T00:01:38Z

Integration

← Older revision		Revision as of 00:01, 13 February 2012
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	<math>\int \int \tan^2x \, dx \, dx = \int (\tan x - x + C)\, dx = -\ln\|\cos x\| - x^2/2 + Cx + C_0</math>		<math>\int \int \tan^2x \, dx \, dx = \int (\tan x - x + C)\, dx = -\ln\|\cos x\| - x^2/2 + Cx + C_0</math>

			===Higher antiderivatives===

			It is not possible to compute higher antiderivatives in terms of [[elementary function]]s, but we can compute them using the [[polylogarithm]].

			===Integration of products with polynomials===

			Using [[integration by parts]], we know that if it is possible to integrate a function <math>k</math> times, it is also possible to use that information to integrate <math>x^{k-1}</math> times the function. Thus, from the above, we can integrate <math>x\tan^2x</math> within elementary functions. However, it is not possible to integrate <math>x^2\tan^2x</math> within elementary functions.

			Explicitly, we have:

			<math>\int x\tan^2x \, dx = x \int \tan^2x \, dx - \int \int \tan^2x \, dx \, dx = x(\tan x - x) + \ln\|\cos x\| + x^2/2 + C = x\tan x + \ln\|\cos x\| - x^2/2 + C</math>

Vipul at 22:48, 12 February 2012

2012-02-12T22:48:40Z

← Older revision		Revision as of 22:48, 12 February 2012
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	\| interval description based on increase/decrease and concave up/down \|\| For each integer <math>n</math>:<br>decreasing and concave up from <math>n\pi - \pi/2</math> to <math>n\pi</math><br>increasing and concave up from <math>n\pi</math> to <math>n\pi + \pi/2</math>.		\| interval description based on increase/decrease and concave up/down \|\| For each integer <math>n</math>:<br>decreasing and concave up from <math>n\pi - \pi/2</math> to <math>n\pi</math><br>increasing and concave up from <math>n\pi</math> to <math>n\pi + \pi/2</math>.
	\|}		\|}

			==Differentiation==

			===First derivative===

			The first derivative can be computed by combining the [[chain rule for differentiation]] and knowledge of the derivatives of the [[square function]] and the [[tangent function]]:

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

			==Integration==

			===First antiderivative===

			We use the identity:

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

			Using this, we rewrite:

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

			where we use that the [[tangent function]] is an antiderivative for the [[secant-squared function]]

			===Second antiderivative===

			We can antidifferentiate the function one more time:

			<math>\int \int \tan^2x \, dx \, dx = \int (\tan x - x + C)\, dx = -\ln\|\cos x\| - x^2/2 + Cx + C_0</math>

Vipul: Created page with "{{particular function}} {{angular function radian convention}} ==Definition== This function is defined as the composite of ..."

2011-08-26T13:59:17Z

Created page with "{{particular function}} {{angular function radian convention}} ==Definition== This function is defined as the composite of ..."

New page

{{particular function}}

{{angular function radian convention}}

==Definition==

This function is defined as the [[defining ingredient::composite of two functions|composite]] of the [[defining ingredient::square function]] and the [[defining ingredient::tangent function]]. Explicitly, it is the function:

<math>x \mapsto (\tan x)^2</math>

<math>(\tan x)^2</math> is written as <math>\tan^2x</math> for shorthand.

==Key data==

{| class="sortable" border="1"
! Item !! Value
|-
|Default [[domain]] || all real numbers ''except'' the odd integer multiples of <math>\pi/2</math>.
|-
|[[range]] || <math>[0,\infty)</math>, i.e., <math>\{ y \mid y \ge 0 \}</math>. All nonnegative reals.<br>no absolute maximum value; absolute minimum value of 0
|-
| [[period]] || <math>\pi</math>, i.e., <math>180\,^\circ</math>.
|-
| [[local maximum value]]s and points of attainment || No local maximum values
|-
| [[local minimum value]]s and points of attainment || 0 at all integer multiples of <math>\pi</math>.
|-
| [[point of inflection|points of inflection]] (both coordinates) || None
|-
| [[vertical asymptote]]s || at all odd multiples of <math>\pi/2</math>, with the function going to <math>+\infty</math> from both directions in each case.
|-
| [[derivative]] || <math>x \mapsto 2\tan x \sec^2x</math>
|-
| [[antiderivative]] || <math>x \mapsto \tan x - x + C</math>. Note that the <math>C</math> value must be constant within each interval between consecutive odd multiples of <math>\pi/2</math>, but can be different in different intervals. The domain overall is not connected.
|-
| interval description based on increase/decrease and concave up/down || For each integer <math>n</math>:<br>decreasing and concave up from <math>n\pi - \pi/2</math> to <math>n\pi</math><br>increasing and concave up from <math>n\pi</math> to <math>n\pi + \pi/2</math>.
|}