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Apparently the sound holes weren't f-shaped to begin with, but c-shaped.     Vielle, reconstruction from a painting by Hans Memling (ca. 1430–1494)Later they became f-shaped.          Gaudenzio Ferrari: Consort of the Viola da braccio Family, 1535Source: Violin, Baroque violin, history, description, viola da gamba, Jose Vazquez, Orpheon ConsortPerhaps this explanation in this post is correct, http://www.maestronet.com/forum/index.php?/topic/315944-f-hole-history-and-reasons/Right! I think I found what I was looking for.  During the period of history in question 1500-1550 the Renaissance was in full swing in France and the world of letters and arts dominated the figure of Francis I who reigned from 1515 to 1546. During his time the cultural life of the country was transformed and many of the greatest artisans of Italy, including Learnado da Vinci, found their way to the court. Music, musicians and instruments were also in demand.It is suggested that instruments commissioned for the French royal court were decorated with the motive "f" in homage to the King and this became the pattern and the norm. (Andrea Amati received a similar commisiion for a large number of instruments for the Spanish court)(cf. Paolo Peterlongo, The Violin. N.B. the author proposes tantalisingly that the suggestion was made by da Vinci himself, a theory he himself dismisses.)Personally, I find all this a bit implausible but for what it's worth...... I think rather, as Magnus has noted, they form part of a sofsicated acoustic and structural system combined with a sense aesthetic beauty.In any case the violin sound holes are called f-holes.On the other hand, the integral sign is a capital S, which used to be long and thin instead of curled up like it is now.From Cajori's History of Mathematical Notation, volume II, page 242Although integral signs and violin sound holes look similar, it's just a coincidence.

The given integral,[math]I'=\displaystyle\int_{0}^{\infty}\dfrac{\cos (x)}{x^{2}+1}\,\mathrm dx[/math]can be evaluated with the help of complex analysis.[math]I=\frac{1}{2}\displaystyle\int_{-\infty}^{\infty}\dfrac{cos(x)}{x^{2}+1}\,\mathrm dx[/math] (Since it is even function so we changed the limits from minus infinity to plus infinity and halfed the integral.)Writing [math]cos(x)=e^{ix}[/math] ,[math]I=\frac{1}{2}\displaystyle\int_{-\infty}^{\infty}\dfrac{e^{i x}}{x^{2}+1}\,\mathrm dx[/math][math]I’[/math] is just the real part of [math]I[/math][math]\implies I=\frac{1}{2}\displaystyle\int_{-\infty}^{\infty}\dfrac{e^{i z}}{z^{2}+1}\,\mathrm dz[/math] (Since now we work in complex plane)Now, we equate[math]z^{2}+1=0[/math][math]\implies z=+i,-i[/math]Here, we take all the points lying in the upper half plane.i.e., [math]z=+i[/math]Now basically finding[math] I [/math] is finding residue of function [math]f(z)=\dfrac{e^{iz}}{z^{2}+1}[/math] at [math]z=+i[/math]So,[math]I=\dfrac{1}{2}(2\pi i) \displaystyle Res_{z = +i}f(z)[/math][math]\implies I=\dfrac{1}{2}(2\pi i)\displaystyle Res_{z = +i}\dfrac{e^{i z}}{z^{2}+1}[/math][math]\implies I=(\pi i)\displaystyle\lim_{z\to +i}(z-i)\dfrac{e^{i z}}{z^{2}+1}[/math][math]\implies I=(\pi i)\displaystyle\lim_{z\to +i}(z-i)\dfrac{e^{iz}}{(z-i)(z+i)}[/math][math]\implies I=(\pi i)\displaystyle\lim_{z\to +i}\dfrac{e^{i z}}{(z+i)}[/math][math]\implies I=(\pi i)\dfrac{e^{i.i}}{(i+i)}[/math][math]\implies I=(\pi i)\dfrac{e^{-1}}{2i}[/math][math]\implies I=(\pi)\dfrac{e^{-1}}{2}[/math][math]\implies I=(\pi)\dfrac{1}{2e}[/math][math]\implies I=\dfrac{\pi}{2e}[/math]So, real part of [math]I[/math] is[math]\dfrac{\pi}{2e} [/math] which is the required answer.Therefore,[math]\displaystyle\int_{0}^{\infty}\dfrac{\cos (x)}{x^{2}+1}\,\mathrm dx=\dfrac{\pi}{2e}[/math]I hope you liked the answer.

[math]\displaystyle \begin {align} \int \frac {xdx}{1+\cos x} & = \int \frac {xdx}{2\cos^2 \left (\frac x2 \right )} \\ \; &= \int x \sec^2 \left (\frac x2 \right)\,d \left (\frac x2 \right) \\ \;& = \int x\,d \left (\tan \left (\frac x2 \right ) \right ) \\ \;&= x\tan \left (\frac x2 \right ) - \int \tan \left (\frac x2 \right )\,dx \\ \; &= x\tan \left (\frac x2 \right ) + 2\int \frac {d\left (\cos \left (\frac x2 \right )\right )}{\cos \left (\frac x2 \right )} \\ \;&= x\tan \left (\frac x2 \right ) + 2\ln \left |\cos \left (\frac x2 \right ) \right | + C \end {align} \tag*{} [/math]

I am going to show you how to write tests for a login page in 3 different ways:without Page Object Modelwith Page Object Modelwith Page FactoryI will do the first one today and add the other 2 in the next few days.So, lets get started.The login page is very simple:You can find it on this demo site:Client Area - PHPTRAVELSThere are multiple tests that can be automated for the login page.I will not do all but the most important 3:login is successfullogin fails when email is incorrectlogin fails when the email is invalidFirst, lets create the test cases.—————————————————LOGIN IS SUCCESSFUL—————————————————type the correct email address in the login fieldtype the correct password address in the password fieldclick the login buttoncheck that “Hello, Alex” is displayed on the welcome pageThe test case uses 2 pages and 4 elements:login pageemail fieldpassword fieldlogin buttonwelcome pagehello elementThe locators for these elements are below.Feel free to try them in Chrome Inspector or Firebug://login page locators private By EMAIL_ID = Domain Premium: By.id("inputEmail"); private By PASSWORD_ID = Domain Premium: By.id("inputPassword"); private By LOGIN_ID = Domain Premium: By.id("login"); //welcome page locators private By HELLO_MESSAGE_ID = By.xpath("//li[@id = 'Secondary_Navbar-Account']/a[@class='dropdown-toggle']"); ——————————————————LOGIN FAILS FOR INCORRECT EMAIL——————————————————type an incorrect email address in the login fieldtype the password address in the password fieldclick the login buttoncheck that an error message is displayed about incorrect login detailsThe test case uses 1 page and 4 elements:login pageemail fieldpassword fieldlogin buttonerror messageThe locators for the first 3 fields are the same as above.The only new locator is for the error message:private By INCORRECT_LOGIN_ERROR_ID = By.xpath("//div[@class = 'alert alert-danger text-center']"); ———————————————————LOGIN FAILS FOR INVALID EMAIL———————————————————type an invalid email address in the login fieldtype the password address in the password fieldclick the login buttoncheck that an error message is displayed about the email address being invalidThe error message does not have a locator as it does not exist in the page.……………………………………………………………The code follows.The test class uses the following methods:setUp(): creates the driver objecttearDown(): closes the driver objectthe test methodsThe code should be pretty straightforward.It is complete so you can play with it by yourself.import static org.junit.Assert.*; import org.junit.After; import org.junit.Before; import org.junit.Ignore; import org.junit.Test; import http://org.openqa.selenium.By; import org.openqa.selenium.JavascriptExecutor; import org.openqa.selenium.WebDriver; import org.openqa.selenium.WebElement; import org.openqa.selenium.chrome.ChromeDriver; import org.openqa.selenium.interactions.Actions; public class LoginPageTests { private final String HOME_TITLE = "Client Area - PHPTRAVELS"; private final String HOME_URL = "Client Area - PHPTRAVELS"; private By EMAIL_ID = By.id("inputEmail"); private By PASSWORD_ID = By.id("inputPassword"); private By LOGIN_ID = By.id("login"); private By INCORRECT_LOGIN_ERROR_ID = By.xpath("//div[@class = 'alert alert-danger text-center']"); private String incorrectDetailsMessage = "Login Details Incorrect. Please try again."; private By HELLO_MESSAGE_ID = By.xpath("//li[@id = 'Secondary_Navbar-Account']/a[@class='dropdown-toggle']"); private String helloMessage = "Hello, %s!"; private String emailValidationError = "Please include an '@' in the email address. '%s' is missing an '@'."; WebDriver driver; JavascriptExecutor jsExecutor; @Before public void setUp() { System.setProperty("webdriver.chrome.driver", "c:/selenium/chromedriver.exe"); driver = new ChromeDriver(); jsExecutor = (JavascriptExecutor) driver; } @After public void tearDown() { driver.quit(); } @Test public void loginWorks() { //open the page driver.get(HOME_URL); assertEquals(HOME_TITLE, driver.getTitle()); //type the email WebElement emailField = driver.findElement(EMAIL_ID); emailField.sendKeys("alexsiminiuc3@gmail.com"); //type the password WebElement passwordField = driver.findElement(PASSWORD_ID); passwordField.sendKeys("Password123!"); //click the login button WebElement loginField = driver.findElement(LOGIN_ID); loginField.click(); //check that the login passed WebElement helloMessageElement = driver.findElement (HELLO_MESSAGE_ID); String expectedMessage = String.format(helloMessage, "Alex"); String actualMessage = helloMessageElement.getText(); assertTrue("Hello message is not displayed!", actualMessage.contains(expectedMessage)); } @Test public void loginFailsForIncorrectEmail() { //open the site driver.get(HOME_URL); assertEquals(HOME_TITLE, driver.getTitle()); //type the email WebElement emailField = driver.findElement(EMAIL_ID); emailField.sendKeys("alexsiminiuc2@gmail.com"); //type the password WebElement passwordField = driver.findElement(PASSWORD_ID); passwordField.sendKeys("Password123!"); //click the login button WebElement loginField = driver.findElement(LOGIN_ID); loginField.click(); //check that the login failed WebElement errorMessageElement = driver.findElement (INCORRECT_LOGIN_ERROR_ID); String actualMessage = errorMessageElement.getText().trim(); assertTrue("Incorrect Details Message is not displayed!", actualMessage.equalsIgnoreCase(incorrectDetailsMessage)); } @Test public void invalidEmailDoesNotWork() { //open the login page driver.get(HOME_URL); assertEquals(HOME_TITLE, driver.getTitle()); //type the email WebElement emailField = driver.findElement(EMAIL_ID); emailField.sendKeys("alexsiminiuc"); //type the password WebElement passwordField = driver.findElement(PASSWORD_ID); passwordField.sendKeys("Password123!"); //click the login button WebElement loginField = driver.findElement(LOGIN_ID); loginField.click(); //check that the error is displayed as tooltip for email String getTooltip = "function getTooltip() { var tooltip = document.querySelector(\"input[id='inputEmail']\"").validationMessage

}