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A mensagem é bem clara: java.lang.ClassNotFoundException: com.mysql.jdbc.Driver
O que indica que você não tem o jar do driver mysql no classpath do teu projeto.
Você utiliza netbeans ou eclipse para desenvolver?
abr 05, 2017 3:49:04 PM org.apache.tomcat.util.digester.SetPropertiesRule begin
ADVERTÊNCIA: [SetPropertiesRule]{Server/Service/Engine/Host/Context} Setting property ‘source’ to ‘org.eclipse.jst.jee.server:desenweb’ did not find a matching property.
abr 05, 2017 3:49:04 PM org.apache.tomcat.util.digester.SetPropertiesRule begin
ADVERTÊNCIA: [SetPropertiesRule]{Server/Service/Engine/Host/Context} Setting property ‘source’ to ‘org.eclipse.jst.jee.server:crud_jsp’ did not find a matching property.
abr 05, 2017 3:49:04 PM org.apache.tomcat.util.digester.SetPropertiesRule begin
ADVERTÊNCIA: [SetPropertiesRule]{Server/Service/Engine/Host/Context} Setting property ‘source’ to ‘org.eclipse.jst.jee.server:jsp’ did not find a matching property.
abr 05, 2017 3:49:04 PM org.apache.catalina.startup.VersionLoggerListener log
INFORMAÇÕES: Server version: Apache Tomcat/8.0.22
abr 05, 2017 3:49:04 PM org.apache.catalina.startup.VersionLoggerListener log
INFORMAÇÕES: Server built: Apr 29 2015 11:46:15 UTC
abr 05, 2017 3:49:04 PM org.apache.catalina.startup.VersionLoggerListener log
INFORMAÇÕES: Server number: 8.0.22.0
abr 05, 2017 3:49:04 PM org.apache.catalina.startup.VersionLoggerListener log
INFORMAÇÕES: OS Name: Windows 10
abr 05, 2017 3:49:04 PM org.apache.catalina.startup.VersionLoggerListener log
INFORMAÇÕES: OS Version: 10.0
abr 05, 2017 3:49:04 PM org.apache.catalina.startup.VersionLoggerListener log
INFORMAÇÕES: Architecture: x86
abr 05, 2017 3:49:04 PM org.apache.catalina.startup.VersionLoggerListener log
INFORMAÇÕES: Java Home: C:\Program Files\Java\jre1.8.0_121
abr 05, 2017 3:49:04 PM org.apache.catalina.startup.VersionLoggerListener log
INFORMAÇÕES: JVM Version: 1.8.0_121-b13
abr 05, 2017 3:49:04 PM org.apache.catalina.startup.VersionLoggerListener log
INFORMAÇÕES: JVM Vendor: Oracle Corporation
abr 05, 2017 3:49:04 PM org.apache.catalina.startup.VersionLoggerListener log
INFORMAÇÕES: CATALINA_BASE: C:\Users\cristiano\workspace.metadata.plugins\org.eclipse.wst.server.core\tmp0
abr 05, 2017 3:49:04 PM org.apache.catalina.startup.VersionLoggerListener log
INFORMAÇÕES: CATALINA_HOME: C:\Users\cristiano\Downloads\apache-tomcat-8.0.22
abr 05, 2017 3:49:04 PM org.apache.catalina.startup.VersionLoggerListener log
INFORMAÇÕES: Command line argument: -Dcatalina.base=C:\Users\cristiano\workspace.metadata.plugins\org.eclipse.wst.server.core\tmp0
abr 05, 2017 3:49:04 PM org.apache.catalina.startup.VersionLoggerListener log
INFORMAÇÕES: Command line argument: -Dcatalina.home=C:\Users\cristiano\Downloads\apache-tomcat-8.0.22
abr 05, 2017 3:49:04 PM org.apache.catalina.startup.VersionLoggerListener log
INFORMAÇÕES: Command line argument: -Dwtp.deploy=C:\Users\cristiano\workspace.metadata.plugins\org.eclipse.wst.server.core\tmp0\wtpwebapps
abr 05, 2017 3:49:04 PM org.apache.catalina.startup.VersionLoggerListener log
INFORMAÇÕES: Command line argument: -Djava.endorsed.dirs=C:\Users\cristiano\Downloads\apache-tomcat-8.0.22\endorsed
abr 05, 2017 3:49:04 PM org.apache.catalina.startup.VersionLoggerListener log
INFORMAÇÕES: Command line argument: -Dfile.encoding=Cp1252
abr 05, 2017 3:49:04 PM org.apache.catalina.core.AprLifecycleListener lifecycleEvent
INFORMAÇÕES: The APR based Apache Tomcat Native library which allows optimal performance in production environments was not found on the java.library.path: C:\Program Files\Java\jre1.8.0_121\bin;C:\WINDOWS\Sun\Java\bin;C:\WINDOWS\system32;C:\WINDOWS;C:/Program Files/Java/jre1.8.0_121/bin/client;C:/Program Files/Java/jre1.8.0_121/bin;C:/Program Files/Java/jre1.8.0_121/lib/i386;C:\ProgramData\Oracle\Java\javapath;C:\WINDOWS\system32;C:\WINDOWS;C:\WINDOWS\System32\Wbem;C:\WINDOWS\System32\WindowsPowerShell\v1.0;C:\Program Files\MySQL\MySQL Utilities 1.6;C:\Users\cristiano\AppData\Local\Microsoft\WindowsApps;;C:\Program Files\eclipse;;.
abr 05, 2017 3:49:05 PM org.apache.coyote.AbstractProtocol init
INFORMAÇÕES: Initializing ProtocolHandler [“http-nio-3030”]
abr 05, 2017 3:49:06 PM org.apache.tomcat.util.net.NioSelectorPool getSharedSelector
INFORMAÇÕES: Using a shared selector for servlet write/read
abr 05, 2017 3:49:06 PM org.apache.coyote.AbstractProtocol init
INFORMAÇÕES: Initializing ProtocolHandler [“ajp-nio-8009”]
abr 05, 2017 3:49:06 PM org.apache.tomcat.util.net.NioSelectorPool getSharedSelector
INFORMAÇÕES: Using a shared selector for servlet write/read
abr 05, 2017 3:49:06 PM org.apache.catalina.startup.Catalina load
INFORMAÇÕES: Initialization processed in 4948 ms
abr 05, 2017 3:49:06 PM org.apache.catalina.core.StandardService startInternal
INFORMAÇÕES: Starting service Catalina
abr 05, 2017 3:49:06 PM org.apache.catalina.core.StandardEngine startInternal
INFORMAÇÕES: Starting Servlet Engine: Apache Tomcat/8.0.22
abr 05, 2017 3:49:09 PM org.apache.jasper.servlet.TldScanner scanJars
INFORMAÇÕES: At least one JAR was scanned for TLDs yet contained no TLDs. Enable debug logging for this logger for a complete list of JARs that were scanned but no TLDs were found in them. Skipping unneeded JARs during scanning can improve startup time and JSP compilation time.
abr 05, 2017 3:49:09 PM org.apache.coyote.AbstractProtocol start
INFORMAÇÕES: Starting ProtocolHandler [“http-nio-3030”]
abr 05, 2017 3:49:09 PM org.apache.coyote.AbstractProtocol start
INFORMAÇÕES: Starting ProtocolHandler [“ajp-nio-8009”]
abr 05, 2017 3:49:09 PM org.apache.catalina.startup.Catalina start
INFORMAÇÕES: Server startup in 3045 ms
java.lang.ClassNotFoundException: com.mysql.jdbc.Driver
at org.apache.catalina.loader.WebappClassLoaderBase.loadClass(WebappClassLoaderBase.java:1305)
at org.apache.catalina.loader.WebappClassLoaderBase.loadClass(WebappClassLoaderBase.java:1157)
at java.lang.Class.forName0(Native Method)
at java.lang.Class.forName(Unknown Source)
at persistence.Dao.open(Dao.java:15)
at persistence.UsuarioDao.create(UsuarioDao.java:11)
at control.Controle.create(Controle.java:48)
at control.Controle.doPost(Controle.java:35)
at javax.servlet.http.HttpServlet.service(HttpServlet.java:648)
at javax.servlet.http.HttpServlet.service(HttpServlet.java:729)
at org.apache.catalina.core.ApplicationFilterChain.internalDoFilter(ApplicationFilterChain.java:291)
at org.apache.catalina.core.ApplicationFilterChain.doFilter(ApplicationFilterChain.java:206)
at org.apache.tomcat.websocket.server.WsFilter.doFilter(WsFilter.java:52)
at org.apache.catalina.core.ApplicationFilterChain.internalDoFilter(ApplicationFilterChain.java:239)
at org.apache.catalina.core.ApplicationFilterChain.doFilter(ApplicationFilterChain.java:206)
at org.apache.catalina.core.StandardWrapperValve.invoke(StandardWrapperValve.java:219)
at org.apache.catalina.core.StandardContextValve.invoke(StandardContextValve.java:106)
at org.apache.catalina.authenticator.AuthenticatorBase.invoke(AuthenticatorBase.java:502)
at org.apache.catalina.core.StandardHostValve.invoke(StandardHostValve.java:142)
at org.apache.catalina.valves.ErrorReportValve.invoke(ErrorReportValve.java:79)
at org.apache.catalina.valves.AbstractAccessLogValve.invoke(AbstractAccessLogValve.java:617)
at org.apache.catalina.core.StandardEngineValve.invoke(StandardEngineValve.java:88)
at org.apache.catalina.connector.CoyoteAdapter.service(CoyoteAdapter.java:518)
at org.apache.coyote.http11.AbstractHttp11Processor.process(AbstractHttp11Processor.java:1091)
at org.apache.coyote.AbstractProtocol$AbstractConnectionHandler.process(AbstractProtocol.java:668)
at org.apache.tomcat.util.net.NioEndpoint$SocketProcessor.doRun(NioEndpoint.java:1521)
at org.apache.tomcat.util.net.NioEndpoint$SocketProcessor.run(NioEndpoint.java:1478)
at java.util.concurrent.ThreadPoolExecutor.runWorker(Unknown Source)
at java.util.concurrent.ThreadPoolExecutor$Worker.run(Unknown Source)
at org.apache.tomcat.util.threads.TaskThread$WrappingRunnable.run(TaskThread.java:61)
at java.lang.Thread.run(Unknown Source)
java.lang.ClassNotFoundException: com.mysql.jdbc.Driver
at org.apache.catalina.loader.WebappClassLoaderBase.loadClass(WebappClassLoaderBase.java:1305)
at org.apache.catalina.loader.WebappClassLoaderBase.loadClass(WebappClassLoaderBase.java:1157)
at java.lang.Class.forName0(Native Method)
at java.lang.Class.forName(Unknown Source)
at persistence.Dao.open(Dao.java:15)
at persistence.UsuarioDao.create(UsuarioDao.java:11)
at control.Controle.create(Controle.java:48)
at control.Controle.doPost(Controle.java:35)
at javax.servlet.http.HttpServlet.service(HttpServlet.java:648)
at javax.servlet.http.HttpServlet.service(HttpServlet.java:729)
at org.apache.catalina.core.ApplicationFilterChain.internalDoFilter(ApplicationFilterChain.java:291)
at org.apache.catalina.core.ApplicationFilterChain.doFilter(ApplicationFilterChain.java:206)
at org.apache.tomcat.websocket.server.WsFilter.doFilter(WsFilter.java:52)
at org.apache.catalina.core.ApplicationFilterChain.internalDoFilter(ApplicationFilterChain.java:239)
at org.apache.catalina.core.ApplicationFilterChain.doFilter(ApplicationFilterChain.java:206)
at org.apache.catalina.core.StandardWrapperValve.invoke(StandardWrapperValve.java:219)
at org.apache.catalina.core.StandardContextValve.invoke(StandardContextValve.java:106)
at org.apache.catalina.authenticator.AuthenticatorBase.invoke(AuthenticatorBase.java:502)
at org.apache.catalina.core.StandardHostValve.invoke(StandardHostValve.java:142)
at org.apache.catalina.valves.ErrorReportValve.invoke(ErrorReportValve.java:79)
at org.apache.catalina.valves.AbstractAccessLogValve.invoke(AbstractAccessLogValve.java:617)
at org.apache.catalina.core.StandardEngineValve.invoke(StandardEngineValve.java:88)
at org.apache.catalina.connector.CoyoteAdapter.service(CoyoteAdapter.java:518)
at org.apache.coyote.http11.AbstractHttp11Processor.process(AbstractHttp11Processor.java:1091)
at org.apache.coyote.AbstractProtocol$AbstractConnectionHandler.process(AbstractProtocol.java:668)
at org.apache.tomcat.util.net.NioEndpoint$SocketProcessor.doRun(NioEndpoint.java:1521)
at org.apache.tomcat.util.net.NioEndpoint$SocketProcessor.run(NioEndpoint.java:1478)
at java.util.concurrent.ThreadPoolExecutor.runWorker(Unknown Source)
at java.util.concurrent.ThreadPoolExecutor$Worker.run(Unknown Source)
at org.apache.tomcat.util.threads.TaskThread$WrappingRunnable.run(TaskThread.java:61)
at java.lang.Thread.run(Unknown Source)
java.lang.ClassNotFoundException: com.mysql.jdbc.Driver
at org.apache.catalina.loader.WebappClassLoaderBase.loadClass(WebappClassLoaderBase.java:1305)
at org.apache.catalina.loader.WebappClassLoaderBase.loadClass(WebappClassLoaderBase.java:1157)
at java.lang.Class.forName0(Native Method)
at java.lang.Class.forName(Unknown Source)
at persistence.Dao.open(Dao.java:15)
at persistence.UsuarioDao.create(UsuarioDao.java:11)
at control.Controle.create(Controle.java:48)
at control.Controle.doPost(Controle.java:35)
at javax.servlet.http.HttpServlet.service(HttpServlet.java:648)
at javax.servlet.http.HttpServlet.service(HttpServlet.java:729)
at org.apache.catalina.core.ApplicationFilterChain.internalDoFilter(ApplicationFilterChain.java:291)
at org.apache.catalina.core.ApplicationFilterChain.doFilter(ApplicationFilterChain.java:206)
at org.apache.tomcat.websocket.server.WsFilter.doFilter(WsFilter.java:52)
at org.apache.catalina.core.ApplicationFilterChain.internalDoFilter(ApplicationFilterChain.java:239)
at org.apache.catalina.core.ApplicationFilterChain.doFilter(ApplicationFilterChain.java:206)
at org.apache.catalina.core.StandardWrapperValve.invoke(StandardWrapperValve.java:219)
at org.apache.catalina.core.StandardContextValve.invoke(StandardContextValve.java:106)
at org.apache.catalina.authenticator.AuthenticatorBase.invoke(AuthenticatorBase.java:502)
at org.apache.catalina.core.StandardHostValve.invoke(StandardHostValve.java:142)
at org.apache.catalina.valves.ErrorReportValve.invoke(ErrorReportValve.java:79)
at org.apache.catalina.valves.AbstractAccessLogValve.invoke(AbstractAccessLogValve.java:617)
at org.apache.catalina.core.StandardEngineValve.invoke(StandardEngineValve.java:88)
at org.apache.catalina.connector.CoyoteAdapter.service(CoyoteAdapter.java:518)
at org.apache.coyote.http11.AbstractHttp11Processor.process(AbstractHttp11Processor.java:1091)
at org.apache.coyote.AbstractProtocol$AbstractConnectionHandler.process(AbstractProtocol.java:668)
at org.apache.tomcat.util.net.NioEndpoint$SocketProcessor.doRun(NioEndpoint.java:1521)
at org.apache.tomcat.util.net.NioEndpoint$SocketProcessor.run(NioEndpoint.java:1478)
at java.util.concurrent.ThreadPoolExecutor.runWorker(Unknown Source)
at java.util.concurrent.ThreadPoolExecutor$Worker.run(Unknown Source)
at org.apache.tomcat.util.threads.TaskThread$WrappingRunnable.run(TaskThread.java:61)
at java.lang.Thread.run(Unknown Source)
Vamos com calma.
Adicionar o jar pelo build path é fundamental para que você consiga desenvolver utilizando o eclipse.
Mas, quando você executa o projeto ou o exporta para colocar em um servidor, é preciso que os jars que utilizou estejam disponíveis para o jar ou war gerados.
No teu caso, você não precisaria adicionar o jar no build path, basta adicionar o mesmo à pasta lib, dentro de WEB-INF e isso resolverá a questão do erro na execução.
Com a adição de um jar na pasta WEB-INF/lib, o eclipse, automaticamente, adiciona o jar no build path.