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 362456 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.0757 1211848 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0757 1211848 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1477 4115248 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1491 4142832 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1500 4160544 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1502 4191576 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1512 4214296 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1512 4214296 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.2090 4849224 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.2118 4950208 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.2124 4971136 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.2216 5043616 13. Joomla\CMS\Plugin\PluginHelper::importPlugin() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:189 0.2216 5043872 14. Joomla\CMS\Plugin\PluginHelper::import() /var/www/astarmathsandphysics/libraries/src/Plugin/PluginHelper.php:182

Integration Using Two Substitutions

If an integral can be evaluated using the substitution method, it of often simpler to make two substitutions. The first seeks to simplify the integrand into a more familiar form which.

Example:

Complete the square inside the square root to obtain

Now substituteThe integral becomes

Now substituteThe integral becomes

This can be evaluated using the trigonometric identityrearranged asWe have

Now use the substitutions to obtain the result in terms of x. Useand

Now use u=x+1 to obtain the result

Example:

Complete the square inside the square root to obtain

Now substituteThe integral becomes

Now substituteThe integral becomes

Useandsuccessively to obtain