I don't have a strong background in electronics, so this task is challenging for me. I need to build a circuit that will get an output like this. In my circuit, I can only use voltage sources, resistors, capacitors, and transistors. The max. total capacitance is limited to 10pF, the total resistance is limited to 100kOhm. The smallest size transistor is 0.4um/0.35um.
Based on the transistor sizes i presume this is IC level stuff. If so start with a voltage source / resistor as a reference current. Then use 2 (or more) transistors as a current mirror. Use that mirrored current to charge your sampling capacitor with a fixed current and your get a linear slope.
Your input voltage would then be operating a transistor that would short out that integrating capacitance to keep its voltage near 0. And once the input rises lets the integrator run up.
You might throw some more transistors at the problem say to disable the current mirror when the input is low. But this would be the majority of it.