We also know that if F and G are antiderivatives off and g, respectively, then F+ G is an antiderivative of f + g. Thus, with simplification, an antiderivative of f(x) = 9x9 – 3x6 + 15x3 is 9/10 9/10 x 10 3/7 7 + 15/4 15/4 x4 Step 5 Finally, if F is an antiderivative off and C is any constant, then F + C is also an anti-derivative of f. If a value for C is not specified, then F + C is the most general form of an anti-derivative for f. Putting all this together then, and simplifying where possible, we have that the most general antiderivative of f(x) = 9x9 – 3x6 + 15x3 is F(x) = C+ 5r10 5 7 3x 7 + 4 15x 4 x Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places. 56 6. sin x dx, n = 4 1.5639 X Find the most general antiderivative of the function. (Check your answer by differentiation. Use C for the constant of the antiderivative.) F(x) = 2 + 2 - 3 3 = 4 43 F(x) = IC- + 4x 3 x 5 + + 에 4
