Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; plgContentJComments has a deprecated constructor in /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php on line 25 Call Stack: 0.0001 362392 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.0473 1211736 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0473 1211736 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1164 4127288 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1177 4154824 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1184 4172536 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1185 4203056 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1195 4225776 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1195 4225776 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.1639 4907448 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.1674 5099600 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.1679 5120528 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.1782 5321288 13. Joomla\CMS\Plugin\PluginHelper::importPlugin() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:189 0.1782 5321544 14. Joomla\CMS\Plugin\PluginHelper::import() /var/www/astarmathsandphysics/libraries/src/Plugin/PluginHelper.php:182

Improper Integrals

An improper integral is one where either of the following holds

  1. one of the integrals isoror the limits areand

andare all improper integrals.

  1. the integrand (the function being integrated) includes a term evaluated at one or both limits which takes the form

andare all improper integrals. The first of these integrands,includes the factorwhich tends toastends toand which tends to 0 astends to infinity . The second includes the factorsandwhich tend toand 0 respectively astends toThe third includes the termsandwhich tend to 0 andastends to 0.

Improper integrands can often be evaluated because the integrand tends to 0 at the troublesome limit, or if the integrand is of the format one or both limits, one factor tends to 0 faster than the other tends to infinity. This is true for the improper integral tends to zero faster than any power oftends to infinity,so we may write

Having the integrand tend to zero at the limits is not sufficient for the integral to be able to be evaluated. The integrand must tend to 0 fast enough. The integralis not defined becausedoes not tend to 0 fast enough. In fact,