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A little cash and a dash of creativity can result in some dramatic changes.<br><br>Install a lazy Susan in those odd corner cabinets. It can be difficult to properly use the [http://Www.Wonderhowto.com/search/storage+space/ storage space] they provide, if you have to get down on your hands and knees to search for the items in the back corners. A lazy Susan will allow you to reach your items easily every time.<br><br>Windows and doors make a perfect home improvement project. You can add an indoor window box or change out your old patio doors for French doors. It is often possible to find good deals on these types of materials at the home improvement store. You won't have to spend a lot of money to make a dramatic change.<br><br>Refurbishing an unfinished basement floor can bring the space together while increasing the functionality of the basement. Family members and friends will want to go down there more than they did before and one can create a new area to entertain guests. 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Shims go up to about 3/8 of an inch to completely flat.<br><br>A great tip for home improvement is to avoid putting your own personal stamp on improvements. This will make it less likely that you will earn a return on the investment because what looks good to you may not look good to someone else. If you use renovations simply to express yourself, do not consider the renovations as an investment in the resale value of your home.<br><br>When partaking in a large home renovation project, you should aim to have a clear vision of what you want. If the contractor feels like he or she can depend on the plans, things are likely to go much smoother. However, if the contractor does not feel confident in the plans, he or she may be afraid to do anything.<br><br>Remember that is easy to make a home improvement project a lot more complicated than it should be. Take the time to carefully plan your repairs and you will eliminate much grief and frustration! 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The '''[[integral]] of the [[Trigonometric_functions#Reciprocal_functions|secant function]]''' of [[trigonometry]] was the subject of one of the "outstanding open problems of the mid-seventeenth century", solved in 1668 by [[James Gregory (mathematician)|James Gregory]].<ref name="rickey-tuchinsky">V. Frederick Rickey and Philip M. Tuchinsky, "An Application of Geography to Mathematics: History of the Integral of the Secant", ''[[Mathematics Magazine]]'', volume 53, number 3, May 1980, pages 162–166.</ref>  In 1599, [[Edward Wright (mathematician)|Edward Wright]] evaluated the integral by [[numerical method]]s – what today we would call [[Riemann sum]]s.<ref>[[Edward Wright (mathematician)|Edward Wright]], ''Certaine Errors in Navigation, Arising either of the ordinaire erroneous making or vsing of the sea Chart, Compasse, Crosse staffe, and Tables of declination of the Sunne, and fixed Starres detected and corrected'', Valentine Simms, London, 1599.</ref> He wanted the solution for the purposes of [[cartography]] – specifically for constructing an accurate [[Mercator projection]].<ref name="rickey-tuchinsky"/>  In the 1640s, Henry Bond, a teacher of navigation, surveying, and other mathematical topics, compared Wright's numerically computed table of values of the integral of the secant with a table of logarithms of the tangent function, and consequently conjectured<ref name="rickey-tuchinsky"/> that
 
: <math> \int_0^\theta \sec\zeta\,d\zeta = \ln\left|\tan\left(\frac{\theta}{2} + \frac{\pi}{4}\right)\right|. </math>
 
That conjecture became widely known, and in 1665, [[Isaac Newton]] was aware of it.<ref>H. W. Turnbull, editor, ''The Correspondence of Isaac Newton'', Cambridge University Press, 1959–1960, volume 1, pages 13–16 and volume 2, pages 99–100.</ref><ref>[[D. T. Whiteside]], editor, ''The Mathematical Papers of Isaac Newton'', Cambridge University Press, 1967, volume 1, pages 466–467 and 473–475.</ref>
 
The problem was solved by [[Isaac Barrow]]. His proof of the result was the earliest use of [[partial fraction]]s in integration.<ref name="rickey-tuchinsky"/> Adapted to modern notation, Barrow's proof began as follows:
 
: <math>
\int \sec \theta \, d\theta = \int \frac{d\theta}{\cos\theta} = \int \frac{\cos\theta \, d\theta}{\cos^2\theta} = \int \frac{\cos\theta \, d\theta}{1 - \sin^2\theta} = \int \frac{du}{1 - u^2}
</math>
 
This reduces it to the problem of antidifferentiating a [[rational function]] by using partial fractions. The proof goes on from there:
 
: <math>
\begin{align}
\int \frac{du}{1 - u^2} & = \int\frac{du}{(1-u)(1+u)} = \dfrac12\int \left(\frac{1}{1+u} + \frac{1}{1-u}\right)\,du \\[10pt]
& = \frac12 \ln \left|1 + u\right| - \frac12 \ln \left|1 - u\right| + C = \frac12 \ln\left|\frac{1+u}{1-u}\right| + C
\end{align}
</math>
 
Finally, we convert it back to a function of&nbsp;''&theta;'':
 
: <math>
= \left\{\begin{array}{l}
\dfrac12 \ln \left|\dfrac{1+\sin\theta}{1-\sin\theta}\right| + C \\[15pt]
\ln\left|\sec\theta + \tan\theta\right| + C \\[15pt]
\ln\left| \tan\left(\dfrac{\theta}{2} + \dfrac{\pi}{4}\right) \right| + C
\end{array}\right\}\text{ (equivalent forms)}
</math>
 
The third form may be obtained directly by means of the following substitutions.
: <math>
\begin{align}
\sec\theta=\frac{1}{\sin\left(\theta + \dfrac{\pi}{2}\right)}
=\frac{1}{2\sin\left(\dfrac{\theta}{2} + \dfrac{\pi}{4}\right)
\cos\left(\dfrac{\theta}{2} + \dfrac{\pi}{4}\right)}
=\frac{\sec^2\left(\dfrac{\theta}{2} + \dfrac{\pi}{4}\right)}
{2\tan\left(\dfrac{\theta}{2} + \dfrac{\pi}{4}\right)}.
\end{align}
</math>
 
The conventional solution for the [[Mercator projection]] ordinate may be written without the modulus signs since the latitude (&phi;) lies between &minus;&pi;/2 and &pi;/2:
:<math>
y= \ln \tan\!\left(\dfrac{\phi}{2} + \dfrac{\pi}{4}\right).
</math>
 
The problem can also be done by using the [[tangent half-angle substitution]], but the details become somewhat more complicated than in the argument above.
 
==Hyperbolic forms==
Let
:<math>
\begin{align}
  \psi          &=\ln(\sec\theta+\tan\theta),\\
  {\rm e}^\psi  &=\sec\theta+\tan\theta,\\
\sinh\psi      &=\frac12({\rm e}^\psi-{\rm e}^{-\psi})=\tan\theta,\\
\cosh\psi      &=\sqrt{1+\sinh^2\psi}=\sec\theta,\\
\tanh\psi      &=\sin\theta.
  \end{align}
</math>
Therefore
:<math>
\begin{align}
  \int \sec \theta \, d\theta&
    =\tanh^{-1}\! \left(\sin\theta\right)
    =\sinh^{-1}\! \left(\tan\theta\right) 
    =\cosh^{-1}\! \left(\sec\theta\right).
\end{align}
</math>
 
==Gudermannian and lambertian==
:<math>
\begin{align}
\int \sec \theta \, d\theta&  = \mbox{gd}^{-1}(\theta)=\mbox{lam}(\theta).
\end{align}
</math>
gd is the [[Gudermannian function]].
The lambertian form (lam) is encountered in the theory of map projections.<ref name=lee_exact>Lee, L.P. (1976). ''Conformal Projections Based on Elliptic Functions''. Supplement No. 1 to Canadian Cartographer, Vol 13. (Designated as Monograph 16)</ref>
 
 
== Notes and references ==
 
{{reflist}}
 
== See also ==
 
* [[Integral of secant cubed]]
* [[Gudermannian function]]
 
== External links ==
 
* [http://www.jstor.org/stable/2690106 Rickey and Tuchinsky's paper on the history of this integral]
 
[[Category:Integral calculus]]

Latest revision as of 20:53, 11 May 2014

Interested in taking on a home improvement project? This article contains the best of the best tips to help make your job easier. Whether you plan to tackle a major project, or you're looking to change out a few small light fixtures, you'll find some helpful advice by reading the following tricks of the trade.

Make sure that your home is well insulated. Insulation is important in keeping the temperature of your home consistent, regardless of the temperature outside of your home. This is helpful in keeping down the cost of your energy bill in the heating and cooling of your house. Insulation is also effective in reducing the noise coming from outside. Furthermore, it adds to the value of your home (be sure to keep all receipts and invoices so that you can prove what work was done).

Get really good references for contractors you use on your home improvements. It is far too easy for a con artist to take your money and run, leaving you with a wreck of a home and no more money to fix it. Don't trust just anyone. Make sure you have plenty of excellent references, from people you really trust.

Not every home improvement project has to be a major overhaul or a major expense; there are a number of quick and easy updates you can make at a reasonable cost to add value and appeal to your home. Start with paint. Paint your walls, your cabinets, your trim or your doors. You can even paint your stove or floor tiles with specialty paint designed for that purpose. Increase storage space by installing closet or pantry organizers. Replace boring kitchen cabinet knobs with ones that are unique and decorative. While you're at it, do the same with your plain white wall plates. Enhance your new paint job with fresh, bright curtains and sheers. Rent a floor buffer, and buff and wax your hardwood floors. A little cash and a dash of creativity can result in some dramatic changes.

Install a lazy Susan in those odd corner cabinets. It can be difficult to properly use the storage space they provide, if you have to get down on your hands and knees to search for the items in the back corners. A lazy Susan will allow you to reach your items easily every time.

Windows and doors make a perfect home improvement project. You can add an indoor window box or change out your old patio doors for French doors. It is often possible to find good deals on these types of materials at the home improvement store. You won't have to spend a lot of money to make a dramatic change.

Refurbishing an unfinished basement floor can bring the space together while increasing the functionality of the basement. Family members and friends will want to go down there more than they did before and one can create a new area to entertain guests. Putting down a wood, carpet, or synthetic floor will really improve ones basement.

Prep before you paint. Painting the rooms in your home can provide a new face lift for a little bit of nothing if you are prepared. Don't try to do it all in one day or weekend. Be prepared before you start. Have all of the necessary tools on hand.

When you decide to begin a home improvement project, know how much work is required. Take a few minutes to create a detailed list of each and every project that must be done. You should also consider getting a second opinion to make sure you haven't missed important details. If you know exactly what must be done, you will be able to economize.

You will need to purchase shims prior to installing your kitchen cabinets. They are used to keep all of the cabinets level and straight. They are especially helpful if the kitchen floor is not completely level or the wall is bowed. Shims go up to about 3/8 of an inch to completely flat.

A great tip for home improvement is to avoid putting your own personal stamp on improvements. This will make it less likely that you will earn a return on the investment because what looks good to you may not look good to someone else. If you use renovations simply to express yourself, do not consider the renovations as an investment in the resale value of your home.

When partaking in a large home renovation project, you should aim to have a clear vision of what you want. If the contractor feels like he or she can depend on the plans, things are likely to go much smoother. However, if the contractor does not feel confident in the plans, he or she may be afraid to do anything.

Remember that is easy to make a home improvement project a lot more complicated than it should be. Take the time to carefully plan your repairs and you will eliminate much grief and frustration! Practical, no-nonsense information like this will make it easier for you to complete your chosen project on time and within budget.

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