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If you have a piecewise function, you need to separate the integral over that function into the same number of (sub-)intervals: $$\begin{align}\int_0^4x(t)e^{-jk\omega_0t}dt&=\int_0^1x(t)e^{-jk\omega_0t}dt+\int_1^2x(t)e^{-jk\omega_0t}dt+\int_2^4x(t)e^{-jk\omega_0t}dt\\&=\int_0^1(-t)e^{-jk\omega_0t}dt+\int_1^2e^{-jk\omega_0t}dt\end{align}$$ with $\...


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