en.wikipedia.org website review
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SEO Keyword summary for en.wikipedia.org/wiki/trigonometric_substitution
Keywords are extracted from the main content of your website and are the primary indicator of the words this page could rank for. By frequenty count we expect your focus keyword to be theta
Focus keyword
Short and long tail
Short Tail Keywords theta frac displaystyle |
long Tail Keywords (2 words) integrands containing a2 x2 containing a2 sidebar hide trigonometric substitution |
long Tail Keywords (3 words) integrands containing a2 containing a2 x2 examples of case move to sidebar case i integrands a2 x2 toggle a2 x2 subsection |
en.wikipedia.org On-Page SEO Scan
Descriptive Elements
The <head> element of a en.wikipedia.org/wiki/trigonometric_substitution page is used to inform the browser and visitors of the page about the general meta information. The head section of the page is where we place the page title, the definition of the HTML version used, the language of in which the page is written. In the head section we can also include JavaScript and CSS (markup) files for the page.
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trigonometric substitution wikipedia
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wikipedia free encyclopedia displaystyle int abftdtfbfa xasin theta sin cos frac dxsqrt quad dxacos dtheta arcsin beginalignedint acos sqrt ptint pttheta ptarcsin xacendaligned leftfrac xarightbiggl rightarcsin textstyle dfrac leq geq dxint pta rightdtheta ptfrac lefttheta rightc leftarcsin xafrac xasqrt cendaligned rightpi biggl ptleftfrac rightleftfrac rightrightfrac endaligned dxleftfrac right ptleft rightleft rightfrac cdot xatan tan sec dxa dxasec arctan asec aarctan aneq xtan dxsec bigg xbigg ptendaligned dxfrac leftsqrt xaln xarightrightc leftxsqrt xsqrt arightrightcendaligned xasec dxx operatorname arcsec xacdot rightrightc fsinxcosxdxint fleftupm rightduusinx fleftpm urightduucosx fleftfrac rightduutan duint duufrac ctan xasinh dxacosh udu cosh xsinh acosh udusqrt sinh ucosh ptuc ptsinh zoperatorname arsinh zlnzsqrt beginalignedsinh xacln ptln arightcendaligned icon wikimedia foundation powered mediawiki
Mobile SEO en.wikipedia.org/wiki/trigonometric_substitution
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Marketing / lead generation for en.wikipedia.org/wiki/trigonometric_substitution
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Facebook shares | Facebook likes | ||
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SERP Preview
SERP Title
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Domain Level SEO
Domain name
16 characters long
Domain name SEO Impact
Path name
substitution found in path !
trigonometric found in path !
Structured data
Publisher Markup
Other Structured data
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Correct processing of non-existing pages?
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Robots.txt found?
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Navigation and internal links
Navigation
Url seperator
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statistics
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w httpsenwikipediaorgwindexphptitletrigonometricsubstitutionoldid1245631698
page information
printable version
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wiki trigonometry
outline
history
usage
trigonometric function
cos
inverse
generalized trigonometry
identity
exact constants
tables
unit circle
sines
cosines
tangents
cotangents
pythagorean theorem
calculus
inverse functions
derivatives
trigonometric series
hipparchus
ptolemy
brahmagupta
alhasib
albattani
regiomontanus
de moivre
euler
fourier
fundamental theorem
limits
continuity
rolles theorem
mean value theorem
inverse function theorem
differential
derivative
generalizations
differential
infinitesimal
differentiation notation
second derivative
implicit differentiation
logarithmic differentiation
related rates
taylors theorem
rules and identities
product
chain
power
quotient
lhpitals rule
inverse
general leibniz
fa di brunos formula
reynolds
integrals
lists of integrals
integral transform
differentiation under the integral sign
antiderivative
improper integrals
riemann integral
lebesgue integral
contour integration
inverse functions
by parts
washers
shells
substitution
tangent halfangle
euler
eulers formula
partial fractions
heavisides method
changing order
reduction formulas
risch algorithm
series
geometric
arithmeticogeometric
harmonic
alternating
power
binomial
taylor
convergence tests
summand limit term test
ratio
root
integral
direct comparison
limit comparison
alternating series
cauchy condensation
dirichlet
abel
vector
gradient
divergence
curl
laplacian
directional derivative
identities
gradient
greens
stokes
divergence
generalized stokes
helmholtz decomposition
multivariable
matrix
tensor
exterior
geometric
partial derivative
multiple integral
line integral
surface integral
volume integral
jacobian
hessian
calculus on euclidean space
generalized functions
limit of distributions
fractional
malliavin
stochastic
variations
precalculus
history
glossary
list of topics
integration bee
mathematical analysis
nonstandard analysis
mathematics
radical function
integral of secant cubed
integral of the secant function
weierstrass
tangent halfangle formulas
hyperbolic functions
the identity cosh2xsinh2x1displaystyle cosh 2xsinh 2x1
the relation sinh1zarsinhzlnzz21displaystyle sinh 1zoperatorname arsinh zlnzsqrt z21
stewart james
brookscole
isbn
thomas george b
hass joel
addisonwesley
burkill integral
bochner integral
daniell integral
darboux integral
henstockkurzweil integral
haar integral
hellinger integral
khinchin integral
kolmogorov integral
lebesguestieltjes integral
pettis integral
pfeffer integral
riemannstieltjes integral
regulated integral
partial fractions
parametric derivatives
laplace transform
laplaces method
numerical integration
simpsons rule
trapezoidal rule
gaussian integral
dirichlet integral
incomplete
boseeinstein integral
frullani integral
common integrals in quantum field theory
stochastic integrals
it integral
russovallois integral
stratonovich integral
skorokhod integral
basel problem
eulermaclaurin formula
gabriels horn
proof that 227 exceeds
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SEO Advice for en.wikipedia.org
In this section we provide pointers on how you can to optimize your web page so it can be found more easily by search engines and how to make it rank higher by optimizing the content of the page itself. For each of the individual criteria the maximum score is 100%. A score below 70% is considered to be indication that the page is not complying with general SEO standards and should be evaluated and/or fixed. Not every factor is weighted the same and some are not as important as others. Relatively unimportant factors like meta keywords are not included in the overall score.
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PageTitle | 100% | Far too many sites lack a page title. A page title is the first thing that shows in the search results so always use the title element. | |
Title relevance | 87% | A title should reflect the contents of a site. This site has a 67 % match | |
Title Length | 80% | Limit your title to anywhere between 40 and 70 characters. Your title was 39 characters long | |
Meta Description | 0% | A meta description is the second element that shows in the search results so always use the meta description. | |
Meta description length | 0% | The meta description should be between 145 and 160 characters. This meta description is 1 characters long. | |
Meta description relevance | 0% | Meta Description should reflect the contents of a site. This site has a 0 % match | |
Number of internal links | 30% | Linking to internal pages makes pages easier to find for search engines. Try to keep the number of links on your page roughly below 100. There are 259 internal links on this page. | |
Folder structure | 100% | We found a folder structure in the links on your page. A good folder structure makes a site easier to navigate. We found 3 level 1 folders and 8 folders above or in the first level of navigation. | |
Headings | 100% | Headers should reflect the contents of a site. This site has a 50 % match | |
Links | 14% | Link anchors should to some degree reflect the contents of a site. This site has a 7 % match | |
Image alt tags | 42% | Image alt tags should to some degree reflect the contents of a site. This site has a 15 % match | |
Bold and italic | 100% | Bold and italic tags should reflect the contents of a site to some degree. This site has a 44 % match | |
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WordCount | 20% | An ideal page contains between 400 and 600 words.This page contains 3377 words | |
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Gzip compression | 30% | This site does not use Gzip compression. Pages may not display as fast as they could | |
Keywords in Domainname | 30% | There are no important keywords in your domain name | |
Keywords in domain path | 100% | There are important keywords in the domain path | |
Structured Data | 100% | Structured data makes it easier for search engines to index your website | |
Inline css | 0% | Do not use inline css declarations. Inline css will slow down the rendering of the website. We detected 253 inline style declarations ( <a style="color:green">) with a size of 8866 bytes | |
Excessive use of the same words | 100% | There is no indication that there are one or more keywords that are used excessively. | |
Frames or iframes | 100% | Perfect, detected not (i)frames on your webpagina | |
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Css | 30% | We detected too much (2) CSS files on your page. Css files block the loading of a webpage. | |
Javascript | 100% | Perfect, we did not detect too many blocking JavaScript files | |
Mobile Website | 100% | Perfect, we found a responsive design for mobile users | |
Most important heading | 100% | Perfect, we detected a correct use of the most important (h1) heading! | |
Normalized headings | 40% | We dit not font a normalized heading structure. A heading 2 (h2) for example should be followed by a heading of an equal level (h2), a child heading (h3) or even a aprent heading (h1). |
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en.wikipedia.org images and descriptions
99 images found at en.wikipedia.org Images can improve the user experience for a website by making a pag visually appealing Images can also add extra keyword relevance to a webpage by using alt tags. Images can also slow down a website. If the width and height for a picture is not specified for a browser know in advance how large the image is. A browser must first load the picture and see before it knows how much space should be on the page. Upon reservation In the meantime, the browser can do little but wait. When the height and width for the plate are given in the HTML code, a browser just continues to build for a page while the images load in the background.
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6dc8b7727d973d3575d22f781010591f86e20436 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\int {\sqrt {a^{2}-x^{2}}}\,dx&=\int {\sqrt {a^{2}-a^{2}\sin ^{2}\theta }}\,(a\cos \theta )\,d\theta \\[6pt]&=\int {\sqrt {a^{2}(1-\sin ^{2}\theta )}}\,(a\cos \theta )\,d\theta \\[6pt]&=\int {\sqrt {a^{2}(\cos ^{2}\theta )}}\,(a\cos \theta )\,d\theta \\[6pt]&=\int (a\cos \theta )(a\cos \theta )\,d\theta \\[6pt]&=a^{2}\int \cos ^{2}\theta \,d\theta \\[6pt]&=a^{2}\int \left({\frac {1+\cos 2\theta }{2}}\right)\,d\theta \\[6pt]&={\frac {a^{2}}{2}}\left(\theta +{\frac {1}{2}}\sin 2\theta \right)+c\\[6pt]&={\frac {a^{2}}{2}}(\theta +\sin \theta \cos \theta )+c\\[6pt]&={\frac {a^{2}}{2}}\left(\arcsin {\frac {x}{a}}+{\frac {x}{a}}{\sqrt {1-{\frac {x^{2}}{a^{2}}}}}\right)+c\\[6pt]&={\frac {a^{2}}{2}}\arcsin {\frac {x}{a}}+{\frac {x}{2}}{\sqrt {a^{2}-x^{2}}}+c.\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/00648715967922d2740a30e742875ae05a2a32ba height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\int _{-1}^{1}{\sqrt {4-x^{2}}}\,dx&=\int _{-\pi /6}^{\pi /6}{\sqrt {4-4\sin ^{2}\theta }}\,(2\cos \theta )\,d\theta \\[6pt]&=\int _{-\pi /6}^{\pi /6}{\sqrt {4(1-\sin ^{2}\theta )}}\,(2\cos \theta )\,d\theta \\[6pt]&=\int _{-\pi /6}^{\pi /6}{\sqrt {4(\cos ^{2}\theta )}}\,(2\cos \theta )\,d\theta \\[6pt]&=\int _{-\pi /6}^{\pi /6}(2\cos \theta )(2\cos \theta )\,d\theta \\[6pt]&=4\int _{-\pi /6}^{\pi /6}\cos ^{2}\theta \,d\theta \\[6pt]&=4\int _{-\pi /6}^{\pi /6}\left({\frac {1+\cos 2\theta }{2}}\right)\,d\theta \\[6pt]&=2\left[\theta +{\frac {1}{2}}\sin 2\theta \right]_{-\pi /6}^{\pi /6}=[2\theta +\sin 2\theta ]{\biggl |}_{-\pi /6}^{\pi /6}\\[6pt]&=\left({\frac {\pi }{3}}+\sin {\frac {\pi }{3}}\right)-\left(-{\frac {\pi }{3}}+\sin \left(-{\frac {\pi }{3}}\right)\right)={\frac {2\pi }{3}}+{\sqrt {3}}.\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/331bd80b5e0c5a19ece342b80e800bd3d1bc2093 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\int _{-1}^{1}{\sqrt {4-x^{2}}}\,dx&=\left[{\frac {2^{2}}{2}}\arcsin {\frac {x}{2}}+{\frac {x}{2}}{\sqrt {2^{2}-x^{2}}}\right]_{-1}^{1}\\[6pt]&=\left(2\arcsin {\frac {1}{2}}+{\frac {1}{2}}{\sqrt {4-1}}\right)-\left(2\arcsin \left(-{\frac {1}{2}}\right)+{\frac {-1}{2}}{\sqrt {4-1}}\right)\\[6pt]&=\left(2\cdot {\frac {\pi }{6}}+{\frac {\sqrt {3}}{2}}\right)-\left(2\cdot \left(-{\frac {\pi }{6}}\right)-{\frac {\sqrt {3}}{2}}\right)\\[6pt]&={\frac {2\pi }{3}}+{\sqrt {3}}\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1c09e16acbfd7c4c1079c381944b55247f8feada height: height attribute not set width: width attribute not set description: {\displaystyle \int {\frac {dx}{a^{2}+x^{2}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0f4dcc45c93a4a99ec39af3da242c8e3738f281d height: height attribute not set width: width attribute not set description: {\displaystyle x=a\tan \theta ,\quad dx=a\sec ^{2}\theta \,d\theta ,\quad \theta =\arctan {\frac {x}{a}},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1c65e486a1f8cafb8397f72820972c35efacd858 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\int {\frac {dx}{a^{2}+x^{2}}}&=\int {\frac {a\sec ^{2}\theta \,d\theta }{a^{2}+a^{2}\tan ^{2}\theta }}\\[6pt]&=\int {\frac {a\sec ^{2}\theta \,d\theta }{a^{2}(1+\tan ^{2}\theta )}}\\[6pt]&=\int {\frac {a\sec ^{2}\theta \,d\theta }{a^{2}\sec ^{2}\theta }}\\[6pt]&=\int {\frac {d\theta }{a}}\\[6pt]&={\frac {\theta }{a}}+c\\[6pt]&={\frac {1}{a}}\arctan {\frac {x}{a}}+c,\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3c4be4ca48aafe304408dc86889e3c578a7be30f height: height attribute not set width: width attribute not set description: {\displaystyle a\neq 0.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f94a7aa5c2e6590955bae4ee675a32540ac6dbbf height: height attribute not set width: width attribute not set description: {\displaystyle \theta =\arctan {\frac {x}{a}},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0077f2811545527f38a243dc04e3ed4f29ee90a9 height: height attribute not set width: width attribute not set description: {\displaystyle -{\frac {\pi }{2}}<\theta <{\frac {\pi }{2}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/32c9059b8650edc9337a3e331c327e9a5257c8e6 height: height attribute not set width: width attribute not set description: {\displaystyle \int _{0}^{1}{\frac {4\,dx}{1+x^{2}}}\,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0f19cdec4ea2124e171ec88be8c6e25b3876f5ce height: height attribute not set width: width attribute not set description: {\displaystyle x=\tan \theta ,\,dx=\sec ^{2}\theta \,d\theta ,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0394785cea95499ca1b7fd844bd96d3a36a758fe height: height attribute not set width: width attribute not set description: {\displaystyle \theta =\arctan x.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0d614f492a627bff0537366d151afc0449158485 height: height attribute not set width: width attribute not set description: {\displaystyle \arctan 0=0} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4274e4635af772bf71c7ec04b6f58f6c95913432 height: height attribute not set width: width attribute not set description: {\displaystyle \arctan 1=\pi /4,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a1fdc8a13ac2312f87a1c7b36cef5ca23eb89075 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\int _{0}^{1}{\frac {4\,dx}{1+x^{2}}}&=4\int _{0}^{1}{\frac {dx}{1+x^{2}}}\\[6pt]&=4\int _{0}^{\pi /4}{\frac {\sec ^{2}\theta \,d\theta }{1+\tan ^{2}\theta }}\\[6pt]&=4\int _{0}^{\pi /4}{\frac {\sec ^{2}\theta \,d\theta }{\sec ^{2}\theta }}\\[6pt]&=4\int _{0}^{\pi /4}d\theta \\[6pt]&=(4\theta ){\bigg |}_{0}^{\pi /4}=4\left({\frac {\pi }{4}}-0\right)=\pi .\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f6316f33eb7e2b8fb35378d35b8f9e6543f92e2f height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\int _{0}^{1}{\frac {4\,dx}{1+x^{2}}}\,&=4\int _{0}^{1}{\frac {dx}{1+x^{2}}}\\[6pt]&=4\left[{\frac {1}{1}}\arctan {\frac {x}{1}}\right]_{0}^{1}\\[6pt]&=4(\arctan x){\bigg |}_{0}^{1}\\[6pt]&=4(\arctan 1-\arctan 0)\\[6pt]&=4\left({\frac {\pi }{4}}-0\right)=\pi ,\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b803094c2df957dda10a019daa3f7b0b552c54cf height: height attribute not set width: width attribute not set description: {\displaystyle \int {\sqrt {a^{2}+x^{2}}}\,{dx}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/becc82ec3cefef1516128cf00fdce37056c6516f height: height attribute not set width: width attribute not set description: {\displaystyle x=a\tan \theta ,\,dx=a\sec ^{2}\theta \,d\theta ,\,\theta =\arctan {\frac {x}{a}},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/eab1f7bba6b0666150eb635d611fdb3c974d463d height: height attribute not set width: width attribute not set description: {\displaystyle {\sqrt {a^{2}}}=a,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f51c4b01b547571951d473f18ea3271fd436bb0a height: height attribute not set width: width attribute not set description: {\displaystyle -{\frac {\pi }{2}}<\theta <{\frac {\pi }{2}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a9f7bfd32ff012708963e0c02c9a16a3a30b6910 height: height attribute not set width: width attribute not set description: {\displaystyle \sec \theta >0} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2061f20ee2d59543dc78f0e90caade6fbbb22f60 height: height attribute not set width: width attribute not set description: {\displaystyle {\sqrt {\sec ^{2}\theta }}=\sec \theta .} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/108a5f1becea83b5cb41021d81544ff3e1bab889 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\int {\sqrt {a^{2}+x^{2}}}\,dx&=\int {\sqrt {a^{2}+a^{2}\tan ^{2}\theta }}\,(a\sec ^{2}\theta )\,d\theta \\[6pt]&=\int {\sqrt {a^{2}(1+\tan ^{2}\theta )}}\,(a\sec ^{2}\theta )\,d\theta \\[6pt]&=\int {\sqrt {a^{2}\sec ^{2}\theta }}\,(a\sec ^{2}\theta )\,d\theta \\[6pt]&=\int (a\sec \theta )(a\sec ^{2}\theta )\,d\theta \\[6pt]&=a^{2}\int \sec ^{3}\theta \,d\theta .\\[6pt]\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/35b28bc818f9ffcffedfb2e767d2d578c4a3e038 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\int {\sqrt {a^{2}+x^{2}}}\,dx&={\frac {a^{2}}{2}}(\sec \theta \tan \theta +\ln |\sec \theta +\tan \theta |)+c\\[6pt]&={\frac {a^{2}}{2}}\left({\sqrt {1+{\frac {x^{2}}{a^{2}}}}}\cdot {\frac {x}{a}}+\ln \left|{\sqrt {1+{\frac {x^{2}}{a^{2}}}}}+{\frac {x}{a}}\right|\right)+c\\[6pt]&={\frac {1}{2}}\left(x{\sqrt {a^{2}+x^{2}}}+a^{2}\ln \left|{\frac {x+{\sqrt {a^{2}+x^{2}}}}{a}}\right|\right)+c.\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/785f60e2cf2ed6ade08b29acf52b55b928bdbda4 height: height attribute not set width: width attribute not set description: {\displaystyle x=a\sec \theta ,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/eb4b3be9755542c047e4ecf6806046c37b05b628 height: height attribute not set width: width attribute not set description: {\displaystyle \sec ^{2}\theta -1=\tan ^{2}\theta .} |
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https://upload.wikimedia.org/wikipedia/commons/thumb/8/8b/trig_sub_triangle_3.png/220px-trig_sub_triangle_3.png height: 137 width: 220 description: no alt description found |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9b30f75798064140e18d96a88c285fd35808652c height: height attribute not set width: width attribute not set description: {\displaystyle \int {\frac {dx}{x^{2}-a^{2}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b0adc36a27b13417d665fc5e270f00f76aef98dd height: height attribute not set width: width attribute not set description: {\displaystyle \int {\sqrt {x^{2}-a^{2}}}\,dx} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/cb7821f86646643d27738332a5beabc1da05f84b height: height attribute not set width: width attribute not set description: {\displaystyle x=a\sec \theta ,\,dx=a\sec \theta \tan \theta \,d\theta ,\,\theta =\operatorname {arcsec} {\frac {x}{a}},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9a6cfde00f807fbaf2cef9936951e5235bf8aa05 height: height attribute not set width: width attribute not set description: {\displaystyle 0\leq \theta <{\frac {\pi }{2}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2b4c8d8607cfd12cb95feef5a2517f4d8aa82ab6 height: height attribute not set width: width attribute not set description: {\displaystyle x>0,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/15c2f13f96d0999868a4f95b44f639ac945c5464 height: height attribute not set width: width attribute not set description: {\displaystyle \tan \theta \geq 0} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/356559b906bf1011a4b76c057ff92af0d6ad11e9 height: height attribute not set width: width attribute not set description: {\displaystyle {\sqrt {\tan ^{2}\theta }}=\tan \theta .} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/bfa25abbc33d5141fff3eebfe40b132b19709f60 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\int {\sqrt {x^{2}-a^{2}}}\,dx&=\int {\sqrt {a^{2}\sec ^{2}\theta -a^{2}}}\cdot a\sec \theta \tan \theta \,d\theta \\&=\int {\sqrt {a^{2}(\sec ^{2}\theta -1)}}\cdot a\sec \theta \tan \theta \,d\theta \\&=\int {\sqrt {a^{2}\tan ^{2}\theta }}\cdot a\sec \theta \tan \theta \,d\theta \\&=\int a^{2}\sec \theta \tan ^{2}\theta \,d\theta \\&=a^{2}\int (\sec \theta )(\sec ^{2}\theta -1)\,d\theta \\&=a^{2}\int (\sec ^{3}\theta -\sec \theta )\,d\theta .\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b49a89923ca7c3917d0781c1adbedf34ebf975be height: height attribute not set width: width attribute not set description: {\displaystyle (\sec \theta +\tan \theta )} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5d551bea9f1a33df981d45ab8cf11a1443d6da85 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\int {\sqrt {x^{2}-a^{2}}}\,dx&={\frac {a^{2}}{2}}(\sec \theta \tan \theta +\ln |\sec \theta +\tan \theta |)-a^{2}\ln |\sec \theta +\tan \theta |+c\\[6pt]&={\frac {a^{2}}{2}}(\sec \theta \tan \theta -\ln |\sec \theta +\tan \theta |)+c\\[6pt]&={\frac {a^{2}}{2}}\left({\frac {x}{a}}\cdot {\sqrt {{\frac {x^{2}}{a^{2}}}-1}}-\ln \left|{\frac {x}{a}}+{\sqrt {{\frac {x^{2}}{a^{2}}}-1}}\right|\right)+c\\[6pt]&={\frac {1}{2}}\left(x{\sqrt {x^{2}-a^{2}}}-a^{2}\ln \left|{\frac {x+{\sqrt {x^{2}-a^{2}}}}{a}}\right|\right)+c.\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/87b66c9db6a8b7bf055a75b1fb0ae78a39013cf5 height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {\pi }{2}}<\theta \leq \pi ,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1a4dbbf970b2d2863dcab589eafe006f08e727d7 height: height attribute not set width: width attribute not set description: {\displaystyle x<0} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/07aa684f0a6a72e702500bc2ddfbf225412f2f07 height: height attribute not set width: width attribute not set description: {\displaystyle \tan \theta \leq 0,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/50c6a4046564b29ef7e666b5bd1271315db14c68 height: height attribute not set width: width attribute not set description: {\displaystyle {\sqrt {\tan ^{2}\theta }}=-\tan \theta } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/54ec70f26a22afa5b02ffb9893d5e33e4f23edf1 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\int f(\sin(x),\cos(x))\,dx&=\int {\frac {1}{\pm {\sqrt {1-u^{2}}}}}f\left(u,\pm {\sqrt {1-u^{2}}}\right)\,du&&u=\sin(x)\\[6pt]\int f(\sin(x),\cos(x))\,dx&=\int {\frac {1}{\mp {\sqrt {1-u^{2}}}}}f\left(\pm {\sqrt {1-u^{2}}},u\right)\,du&&u=\cos(x)\\[6pt]\int f(\sin(x),\cos(x))\,dx&=\int {\frac {2}{1+u^{2}}}f\left({\frac {2u}{1+u^{2}}},{\frac {1-u^{2}}{1+u^{2}}}\right)\,du&&u=\tan \left({\frac {x}{2}}\right)\\[6pt]\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a49ac1614b4b912a8521f37a3ff4c0aa9af07f78 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\int {\frac {4\cos x}{(1+\cos x)^{3}}}\,dx&=\int {\frac {2}{1+u^{2}}}{\frac {4\left({\frac {1-u^{2}}{1+u^{2}}}\right)}{\left(1+{\frac {1-u^{2}}{1+u^{2}}}\right)^{3}}}\,du=\int (1-u^{2})(1+u^{2})\,du\\&=\int (1-u^{4})\,du=u-{\frac {u^{5}}{5}}+c=\tan {\frac {x}{2}}-{\frac {1}{5}}\tan ^{5}{\frac {x}{2}}+c.\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d1d76fefaa43fcf0721aadf84cbdc9ee5b685136 height: height attribute not set width: width attribute not set description: {\displaystyle 1/{\sqrt {a^{2}+x^{2}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0c81854655379745b3b53e1d2231df0df5f1a408 height: height attribute not set width: width attribute not set description: {\displaystyle x=a\sinh {u}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5fa3a6571b091c319651f1adb9d99e7998e209b7 height: height attribute not set width: width attribute not set description: {\displaystyle dx=a\cosh u\,du} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fc313680e7f9410109548d12e3b8360e4955869a height: height attribute not set width: width attribute not set description: {\displaystyle \cosh ^{2}(x)-\sinh ^{2}(x)=1} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ca1b341ada3943f7c78484e1a58a2c5b20524d70 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\int {\frac {dx}{\sqrt {a^{2}+x^{2}}}}&=\int {\frac {a\cosh u\,du}{\sqrt {a^{2}+a^{2}\sinh ^{2}u}}}\\[6pt]&=\int {\frac {\cosh {u}\,du}{\sqrt {1+\sinh ^{2}{u}}}}\\[6pt]&=\int {\frac {\cosh {u}}{\cosh u}}\,du\\[6pt]&=u+c\\[6pt]&=\sinh ^{-1}{\frac {x}{a}}+c.\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b99cf54108345a4830c965ff0d4b05ad3ca49421 height: height attribute not set width: width attribute not set description: {\displaystyle \sinh ^{-1}{z}=\operatorname {arsinh} {z}=\ln(z+{\sqrt {z^{2}+1}})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/20441614472c3f5e71a8f449e4cef1cddfd2d0dc height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\sinh ^{-1}{\frac {x}{a}}+c&=\ln \left({\frac {x}{a}}+{\sqrt {{\frac {x^{2}}{a^{2}}}+1}}\,\right)+c\\[6pt]&=\ln \left({\frac {x+{\sqrt {x^{2}+a^{2}}}}{a}}\,\right)+c.\end{aligned}}} |
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https://login.wikimedia.org/wiki/special:centralautologin/start?type=1x1 height: 1 width: 1 description: no alt description found |
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http://en.wikipedia.org/static/images/footer/wikimedia-button.svg height: 29 width: 84 description: wikimedia foundation |
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