If you're using an Arduino, it's simpler because the Arduino has PWM built-in. But, if you just need some simple logic and you don't need the Arudino, you can use something cheaper that doesn't require programming. (i.e. A regular non-programmable thermostat doesn't have a "brain" or memory.
You just need a transistor or MOSFET to boost the voltage & current. (I haven't looked at the datasheet, but I assume the LTC6992 also needs a transistor/MOSFET to drive the fan.)
or shuld i use something like integrator/capacitor in the output to get something like this:
No. The inertia of the motor & fan take care of the "smoothing". Plus, the transistor/MOSFET operates more efficiently in switching mode.
The tricky part is the transfer function between temperature and fan speed... You can easily end-up with something that oscillates between fast & slow as the fan continuously adjusts and "hunts" for the right speed as the temperature changes follow. (That's assuming a "feedback" system where the fan is cooling whatever you are measuring the temperature of... Like the temperature of a computer... If you just trying to make the room
feel cooler by turning on a fan when it gets warm, that's not an issue.)
It's if you just turn the fan on or off, or maybe have 2-speeds with a time delay before it goes back to slow or off. (It will still tend to cycle like a heater or refrigerator.)