How many antiderivatives does {{MATH_15}} have? {{CMD_16}} {{CMD_17}} {{CMD_18}}none{{CMD_19}} {{CMD_20}}one{{CMD_21}} {{CMD_22}}infinitely many{{CMD_23}} {{CMD_24}} {{CMD_25}} {{CMD_26}} {{CMD_27}} {{ENV_28}} {{COMMENT_29}} {{COMMENT_30}} {{COMMENT_31}} {{COMMENT_32}} {{CMD_33}} The functions {{MATH_34}}, {{MATH_35}}, {{MATH_36}}, {{MATH_37}} and so on, are all antiderivatives of {{MATH_38}}. It is clear that any function {{MATH_39}}, where {{MATH_40}} is a constant, is an antiderivative of {{MATH_41}}. Why? Because, {{MATH_42}}. Could there exist any other antiderivative of {{MATH_43}}, call it {{MATH_44}}, such that {{MATH_45}} is not a sum of {{MATH_46}} and a constant? In that case, for all {{MATH_47}} we would have {{MATH_48}} {{CMD_49}} Recall: Any function whose derivative is equal to {{MATH_50}} on some interval is equal to a constant on that interval. This means that the function {{MATH_51}}, for some constant {{MATH_52}}. This implies that {{MATH_53}}, for all x. {{CMD_54}}