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 360856 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.0460 1210992 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0460 1210992 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1179 4131544 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1192 4159136 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1199 4176848 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1200 4207368 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1211 4230088 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1211 4230088 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.1714 4899904 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.1748 5092056 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.1754 5112984 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.1863 5327688 13. Joomla\CMS\Plugin\PluginHelper::importPlugin() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:189 0.1863 5327944 14. Joomla\CMS\Plugin\PluginHelper::import() /var/www/astarmathsandphysics/libraries/src/Plugin/PluginHelper.php:182 Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; JCommentsACL has a deprecated constructor in /var/www/astarmathsandphysics/components/com_jcomments/classes/acl.php on line 17 Call Stack: 0.0001 360856 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.0460 1210992 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0460 1210992 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1179 4131544 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1192 4159136 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1199 4176848 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1200 4207368 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1211 4230088 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1211 4230088 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.1714 4899904 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.1748 5092056 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.1754 5112984 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.3099 13748080 13. JEventDispatcher->trigger() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:199 0.3101 13748480 14. plgContentJComments->update() /var/www/astarmathsandphysics/libraries/joomla/event/dispatcher.php:160 0.3101 13748480 15. plgContentJComments->onContentAfterDisplay() /var/www/astarmathsandphysics/libraries/joomla/event/event.php:70 0.3103 13756728 16. plgContentJComments->onAfterDisplayContent() /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php:339 0.3106 13758464 17. JComments::show() /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php:282 0.3112 13791512 18. JCommentsFactory::getACL() /var/www/astarmathsandphysics/components/com_jcomments/jcomments.php:188 0.3112 13791912 19. spl_autoload_call() /var/www/astarmathsandphysics/components/com_jcomments/classes/factory.php:274 0.3113 13791992 20. JLoader::load() /var/www/astarmathsandphysics/components/com_jcomments/classes/factory.php:274

Integration Using Partial Fractions

To integration an expression of the form  
\[\frac{p(x)}{q(x)}\]
  where the degree of  
\[p(x)\]
  is less than the degree of  
\[q(x)\]
  write the expression as partial; fractions as much as possible and then integrate using other methods.
Example:  
\[\int \frac{2x-1}{x^2-4x-5} dx\]

\[\int \frac{2x-1}{x^2-4x-5}=\frac{A}{x+1}+\frac{B}{x-5} \rightarrow 2x-1=A(x-5)+B(x+1)\]

\[x=-1 \rightarrow 2 \times -1-1=A(-1-5)+B(-1+1)=-6A \rightarrow A=\frac{-3}{-6}=\frac{1}{2}\]

\[x=5 \rightarrow 2 \times 5-1=A(5-5)+B(5+1)=6B \rightarrow B=\frac{9}{6}=\frac{3}{2}\]
.
The integral becomes  
\[\int \frac{1}{2(x+1)}+\frac{3}{2(x-5)} dx\]
.
Now we can integrate.
\[\begin{equation} \begin{aligned} \int \frac{1}{2(x+1)}+\frac{3}{2(x-5)} dx &=\frac{1}{2}ln(x+1)+\frac{3}{2} ln (x+5)+c \\ &=\frac{1}{2}(ln(x+1)+3 ln(x-5))+c \\ & =\frac{1}{2} ln((x+1)(x-5)^3)+c \end{aligned} \end{equation}\]

Add comment

Security code
Refresh