I am into web development but I handle small projects and I just want to ask does unit testing/Integration testing done by the developer or for testing there is different department? I mean do I need to learn how to write test case too??

Devs should write their own specs.
QA should write their own specs, too.

Why? They both think about the features in different ways, and will therefore test the same features via different approaches. Better feature coverage that way.

Devs should write the unit tests,

TDD done right means the tests exist before the code.
- I don't do it right, but I know what's right.🤷‍♂️
- I write tests though

QA handle their own tests and automation to follow business usage and to ensure what's been built meets requirements and expectations..

The amount of times tests have saved my ass, I always write tests, I even enjoy writing tests and seeing all the green!
Also if you don't write your tests along with your code and leave it for the next developer, a lot of the context and edge-cases are lost forever until they rear their ugly heads.
Write tests for your own peace of mind if not for anything else.

Sure you must write your own test, to member sure that your code is working as expected... And as @Root told you, QA also write their own test

@Vanegas9090 so conclusion is I can't skip learning of integration testing

@rox79 you wouldn’t skip it