Solución

$$\begin{aligned} \int_C \bold{F}\cdot d\bold{r} &= \int_{C_1} \bold{F}\cdot d\bold{r} + \int_{C_2} \bold{F}\cdot d\bold{r} + \int_{C_3} \bold{F}\cdot d\bold{r} + \int_{C_4} \bold{F}\cdot d\bold{r}\\ &= \int_{C_1} \bold{F}\cdot d\bold{r} + \int_{C_2} \bold{F}\cdot d\bold{r} - \int_{-C_3} \bold{F}\cdot d\bold{r} - \int_{-C_4} \bold{F}\cdot d\bold{r}\\ &= \int_a^b \bold{F}(\bold{r}_1(t)) \cdot \bold{r}_1(t)dt + \int_c^d \bold{F}(\bold{r}_2(t)) \cdot \bold{r}_2(t)dt\\ &\;\;\;\; - \int_a^b \bold{F}(\bold{r}_3(t)) \cdot \bold{r}_3(t)dt- \int_c^d \bold{F}(\bold{r}_4(t)) \cdot \bold{r}_4(t)dt\\ \end{aligned}$$