Don your gardening gloves.
Use your utility knife to cut off the broken section of cactus on an angle, at the nearest joint below the damage.
Let the freshly cut section of cactus dry in a warm place for at least a day before replanting, and for as long as two weeks for large broken sections. This will keep fungal soft rot from forming in your broken section.
Pole a small hole in the base of the cactus section that will be placed in the soil.
Place the cut portion of the broken cactus in dry soil, after you notice a dried callous has formed over the cut. Place about one inch of the broken section into the soil. Avoid watering the new section of cactus for a week after planting it.
Water the cactus lightly after a week, and continue to water it weekly, as new roots start to form.