Have a look at connectionstrings.com for every possible variation - a very handy resource I use all the time

Specifically, you want this format:

Data Source=myServerAddress;Initial Catalog=myDataBase;Integrated Security=SSPI;

This, of course, only works if the domain account in question is the one opening the connection.

There's no easy way to connect with arbitrary credentials - but you can impersonate the user in question and then connect.

This can be a bit of a pain. An alternative if the users are on the local network (or you control their browser config) is to use Kerberos authentication on your site. The pages will be served with the relevant user's permissions - then you can use the connection string above and IIS will connect to the Db with the appropriate credentials for each user. This is particularly useful from a security perspective as the Db is able to audit on a per-user basis, and permissions can be per-user/row/column instead of only per-app.

If you want to use different user account then the logged in user you have two options.
Option 1

You can add the user to Application pool Identity.

For this go to advance setting of application pool and edit the identity to use the user you want.
Option 2

Add this in Web config:

<identity impersonate="true" userName="Domain\User" password="Password" />

And use this connection stirng:

<add name="Name" connectionString="Data source=SqlServer;Initial Catalog=DbName;Integrated security=True" providerName="System.Data.SqlClient"/>

For More Details See: https://msdn.microsoft.com/en-us/library/134ec8tc.aspx

Also found another good article here https://www.codeproject.com/tips/520341/implement-impersonation-in-asp-net

Use integrated security:

Integrated Security=SSPI

Which has a variant:

Trusted_Connection=True

The different connection strings (for a variety of databases) can be found on connectionstrings.com .

With both of these you need to ensure that the application is running under the account you need to login with.

Yes, try this:

Data Source=server;Initial Catalog=database;Integrated Security=SSPI;

This specifies that you wish to use integrated Windows authentication where you were still trying to use SQL Server authentication (even though the username you entered looked like a Windows domain / user account SQL server still treats it as standard SQL Server authentication)

Also take a look at connectionstrings.com

Data Source=myServerAddress;Initial Catalog=myDataBase;Integrated Security=SSPI;

The above is a connection string for Windows Authentication against your SQL Server instance.