Add integration_test to integration tests build shard (#71934)

diff --git a/dev/bots/test.dart b/dev/bots/test.dart
index 10b8bef..cc5b607 100644
--- a/dev/bots/test.dart
+++ b/dev/bots/test.dart
@@ -374,6 +374,7 @@
 /// target app.
 Future<void> _runBuildTests() async {
   final List<FileSystemEntity> exampleDirectories = Directory(path.join(flutterRoot, 'examples')).listSync()
+    ..add(Directory(path.join(flutterRoot, 'packages', 'integration_test', 'example')))
     ..add(Directory(path.join(flutterRoot, 'dev', 'integration_tests', 'non_nullable')))
     ..add(Directory(path.join(flutterRoot, 'dev', 'integration_tests', 'flutter_gallery')));
 
@@ -471,18 +472,6 @@
 }) async {
   assert(Platform.isMacOS);
   print('${green}Testing IPA build$reset for $cyan$relativePathToApplication$reset...');
-  // Install Cocoapods.  We don't have these checked in for the examples,
-  // and build ios doesn't take care of it automatically.
-  final File podfile = File(path.join(flutterRoot, relativePathToApplication, 'ios', 'Podfile'));
-  if (podfile.existsSync()) {
-    await runCommand('pod',
-      <String>['install'],
-      workingDirectory: podfile.parent.path,
-      environment: <String, String>{
-        'LANG': 'en_US.UTF-8',
-      },
-    );
-  }
   await _flutterBuild(relativePathToApplication, 'IPA', 'ios',
     release: release,
     verifyCaching: verifyCaching,