chore(elasticloadbalancingv2-actions): update integ tests (aws#21642)

Updating these tests to use the new integ runner and updating the readme to not have someone's commentary about the test/readme not being complete.


----

### All Submissions:

* [ ] Have you followed the guidelines in our [Contributing guide?](https://github.com/aws/aws-cdk/blob/main/CONTRIBUTING.md)

### Adding new Unconventional Dependencies:

* [ ] This PR adds new unconventional dependencies following the process described [here](https://github.com/aws/aws-cdk/blob/main/CONTRIBUTING.md/#adding-new-unconventional-dependencies)

### New Features

* [ ] Have you added the new feature to an [integration test](https://github.com/aws/aws-cdk/blob/main/INTEGRATION_TESTS.md)?
	* [ ] Did you use `yarn integ` to deploy the infrastructure and generate the snapshot (i.e. `yarn integ` without `--dry-run`)?

*By submitting this pull request, I confirm that my contribution is made under the terms of the Apache-2.0 license*

packages/@aws-cdk/aws-elasticloadbalancingv2-actions/README.md

@@ -19,8 +19,58 @@ read [Prepare to use Amazon
 Cognito](https://docs.aws.amazon.com/elasticloadbalancing/latest/application/listener-authenticate-users.html#cognito-requirements).
 Here's an example:
 
-[Example of using AuthenticateCognitoAction](test/integ.cognito.lit.ts)
+```ts
+const lb = new elbv2.ApplicationLoadBalancer(this, 'LB', {
+  vpc,
+  internetFacing: true,
+});
 
-> NOTE: this example seems incomplete, I was not able to get the redirect back to the
-Load Balancer after authentication working. Would love some pointers on what a full working
-setup actually looks like!
+const userPool = new cognito.UserPool(this, 'UserPool');
+const userPoolClient = new cognito.UserPoolClient(this, 'Client', {
+  userPool,
+
+  // Required minimal configuration for use with an ELB
+  generateSecret: true,
+  authFlows: {
+    userPassword: true,
+  },
+  oAuth: {
+    flows: {
+      authorizationCodeGrant: true,
+    },
+    scopes: [cognito.OAuthScope.EMAIL],
+    callbackUrls: [
+      `https://${lb.loadBalancerDnsName}/oauth2/idpresponse`,
+    ],
+  },
+});
+const cfnClient = userPoolClient.node.defaultChild as cognito.CfnUserPoolClient;
+cfnClient.addPropertyOverride('RefreshTokenValidity', 1);
+cfnClient.addPropertyOverride('SupportedIdentityProviders', ['COGNITO']);
+
+const userPoolDomain = new cognito.UserPoolDomain(this, 'Domain', {
+  userPool,
+  cognitoDomain: {
+    domainPrefix: 'test-cdk-prefix',
+  },
+});
+
+lb.addListener('Listener', {
+  port: 443,
+  certificates: [certificate],
+  defaultAction: new actions.AuthenticateCognitoAction({
+    userPool,
+    userPoolClient,
+    userPoolDomain,
+    next: elbv2.ListenerAction.fixedResponse(200, {
+      contentType: 'text/plain',
+      messageBody: 'Authenticated',
+    }),
+  }),
+});
+
+new CfnOutput(this, 'DNS', {
+  value: lb.loadBalancerDnsName,
+});
+
+```

packages/@aws-cdk/aws-elasticloadbalancingv2-actions/package.json

@@ -76,6 +76,7 @@
     "@aws-cdk/assertions": "0.0.0",
     "@aws-cdk/cdk-build-tools": "0.0.0",
     "@aws-cdk/integ-runner": "0.0.0",
+    "@aws-cdk/integ-tests": "0.0.0",
     "@aws-cdk/pkglint": "0.0.0",
     "@types/jest": "^27.5.2",
     "jest": "^27.5.1"

packages/@aws-cdk/aws-elasticloadbalancingv2-actions/test/cognito.integ.snapshot/cdk.out

@@ -0,0 +1 @@
+{"version":"20.0.0"}