Find an antiderivative with and , given that .

Question

Find an antiderivative  with  and  , given that  .

Quizplus · Accepted Answer

By the second fundamental theorem of calculus, we have that the antiderivative of a function f(x) between the limits a and x is F(x) - F(a), where F(x) is any antiderivative of f(x). Therefore, we can find F(x) by adding the given function to any constant, and then evaluate F(11eb69ff_fe99_c815_af94_ff9ed3ca53f1_TB8190_11) - F(11eb69ff_fe99_c814_af94_8f71eaa33bd5_TB8190_11) to get the correct constant of integration. Doing so, we get:

F(x) = 11eb69ff_fe99_c812_af94_eb7eec1c5c09_TB8190_11x + C

where C is any constant. Plugging in the given limits, we get:

F(11eb69ff_fe99_c815_af94_ff9ed3ca53f1_TB8190_11) - F(11eb69ff_fe99_c814_af94_8f71eaa33bd5_TB8190_11) = 11eb69ff_fe99_ef28_af94_a9e9d8653d42_TB8190_11 - 11eb69ff_fe99_ef28_af94_f7284beb5911_TB8190_11 = 11eb69ff_fe99_ef28_af94_a9e9d8653d42_TB8190_11 - (-11eb69ff_fe99_ef28_af94_f7284beb5911_TB8190_11) = 2(11eb69ff_fe99_ef28_af94_a9e9d8653d42_TB8190_11)

Therefore, the correct antiderivative is:

F(x) = 11eb69ff_fe99_c812_af94_eb7eec1c5c09_TB8190_11x + 11eb69ff_fe99_ef28_af94_a9e9d8653d42_TB8190_11

which corresponds to choice C.

Find an Antiderivative with and , Given