feat: add Context as arg to NewIntegrationApp (#15779)

## Description

Closes: #XXXX

This PR adds `sdk.Context` arg to `NewIntegrationApp`. This allows us to set the context updatable while verifying `begin/end block` logic in integration tests.

---

### Author Checklist

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please add links to any relevant follow up issues.*

I have...

* [ ] included the correct [type prefix](https://github.com/commitizen/conventional-commit-types/blob/v3.0.0/index.json) in the PR title
* [ ] added `!` to the type prefix if API or client breaking change
* [ ] targeted the correct branch (see [PR Targeting](https://github.com/cosmos/cosmos-sdk/blob/main/CONTRIBUTING.md#pr-targeting))
* [ ] provided a link to the relevant issue or specification
* [ ] followed the guidelines for [building modules](https://github.com/cosmos/cosmos-sdk/blob/main/docs/docs/building-modules)
* [ ] included the necessary unit and integration [tests](https://github.com/cosmos/cosmos-sdk/blob/main/CONTRIBUTING.md#testing)
* [ ] added a changelog entry to `CHANGELOG.md`
* [ ] included comments for [documenting Go code](https://blog.golang.org/godoc)
* [ ] updated the relevant documentation or specification
* [ ] reviewed "Files changed" and left comments if necessary
* [ ] confirmed all CI checks have passed

### Reviewers Checklist

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your handle next to the items reviewed if you only reviewed selected items.*

I have...

* [ ] confirmed the correct [type prefix](https://github.com/commitizen/conventional-commit-types/blob/v3.0.0/index.json) in the PR title
* [ ] confirmed `!` in the type prefix if API or client breaking change
* [ ] confirmed all author checklist items have been addressed 
* [ ] reviewed state machine logic
* [ ] reviewed API design and naming
* [ ] reviewed documentation is accurate
* [ ] reviewed tests and test coverage
* [ ] manually tested (if applicable)

testutil/integration/example_test.go

@@ -6,6 +6,10 @@ import (
 
 	"cosmossdk.io/log"
 	storetypes "cosmossdk.io/store/types"
+	cmtproto "github.com/cometbft/cometbft/proto/tendermint/types"
+
+	"github.com/google/go-cmp/cmp"
+
 	"github.com/cosmos/cosmos-sdk/runtime"
 	"github.com/cosmos/cosmos-sdk/testutil/integration"
 	sdk "github.com/cosmos/cosmos-sdk/types"
@@ -17,7 +21,6 @@ import (
 	"github.com/cosmos/cosmos-sdk/x/mint"
 	mintkeeper "github.com/cosmos/cosmos-sdk/x/mint/keeper"
 	minttypes "github.com/cosmos/cosmos-sdk/x/mint/types"
-	"github.com/google/go-cmp/cmp"
 )
 
 // Example shows how to use the integration test framework to test the integration of SDK modules.
@@ -29,6 +32,12 @@ func Example() {
 	keys := storetypes.NewKVStoreKeys(authtypes.StoreKey, minttypes.StoreKey)
 	authority := authtypes.NewModuleAddress("gov").String()
 
+	// replace the logger by testing values in a real test case (e.g. log.NewTestLogger(t))
+	logger := log.NewLogger(io.Discard, log.OutputJSONOption())
+
+	cms := integration.CreateMultiStore(keys, logger)
+	newCtx := sdk.NewContext(cms, cmtproto.Header{}, true, logger)
+
 	accountKeeper := authkeeper.NewAccountKeeper(
 		encodingCfg.Codec,
 		runtime.NewKVStoreService(keys[authtypes.StoreKey]),
@@ -47,9 +56,9 @@ func Example() {
 	mintModule := mint.NewAppModule(encodingCfg.Codec, mintKeeper, accountKeeper, nil, nil)
 
 	// create the application and register all the modules from the previous step
-	// replace the logger by testing values in a real test case (e.g. log.NewTestLogger(t))
 	integrationApp := integration.NewIntegrationApp(
-		log.NewLogger(io.Discard, log.OutputJSONOption()),
+		newCtx,
+		logger,
 		keys,
 		encodingCfg.Codec,
 		authModule, mintModule,
@@ -104,6 +113,12 @@ func Example_oneModule() {
 	keys := storetypes.NewKVStoreKeys(authtypes.StoreKey)
 	authority := authtypes.NewModuleAddress("gov").String()
 
+	// replace the logger by testing values in a real test case (e.g. log.NewTestLogger(t))
+	logger := log.NewLogger(io.Discard)
+
+	cms := integration.CreateMultiStore(keys, logger)
+	newCtx := sdk.NewContext(cms, cmtproto.Header{}, true, logger)
+
 	accountKeeper := authkeeper.NewAccountKeeper(
 		encodingCfg.Codec,
 		runtime.NewKVStoreService(keys[authtypes.StoreKey]),
@@ -117,9 +132,9 @@ func Example_oneModule() {
 	authModule := auth.NewAppModule(encodingCfg.Codec, accountKeeper, authsims.RandomGenesisAccounts, nil)
 
 	// create the application and register all the modules from the previous step
-	// replace the logger by testing values in a real test case (e.g. log.NewTestLogger(t))
 	integrationApp := integration.NewIntegrationApp(
-		log.NewLogger(io.Discard),
+		newCtx,
+		logger,
 		keys,
 		encodingCfg.Codec,
 		authModule,