please assist with the questions below

1. (10 points) Use the Antiderivatives and Definite Integrals Theorem to evaluate the following integrals. (similar to 4.4 #9-18) . Ar dx b. f, 4x3 dx c. f 4x3 dx d. Does f, 4x' dx= [ 4x3 dx - f; 4x3 dx ? (yes or no) 2. (10 points) The velocity of a car after t seconds is (27 -t ) feet/sec. (similar to 4.4 #19- 23) a. Plot the velocity vs time for t = 0. b. How many seconds does it take for the care to come to a stop (velocity = 0)? c. How far does the car travel while coming to a stop? d. How many seconds does it take the car to travel half the distance in part (c)? Hint: Solve for t using the Desmos graphing program.3. (10 points) Verify that F(x) is an antiderivative of the integrand f(x) by differentiating F(x) a. Let F(x) = In(1+x-) Compute f(x) = F'(x). b. Then use Part 2 of the Fundamental Theorem to evaluate the definite integral. (similar to 4.5 #13-33) Let a = 1 and b = 4 in F(a) and F(b) in the following integral. (of (x ) dx = F(b)-F(a) c. Check your answer using the Desmos graphing program. For your first entry, enter "f(x) = - 2x 73 1+x2 For your second entry enter "int 1 4 f(x) dx" Show a screen shot. d. Do your answers from (b) and (c) match? (yes or no) 4. (10 points) Using Leibniz's Rule (see Weekly Summary eq #4), find (similar to 4.5 #57- 68) dx d ( f) (t3 ) dt5. (10 points) Use the change of variable technique to find an antiderivative in terms of x. (4.6 #15-26) S=* [In(x)]2 dx 6. (10 points) Evaluate the definite integral. (similar to 4.6 #27-38) a. So (1 + x2 ) ) (3x + x3 ) + dx b. Check your answer with the Simpson's rule calculator. https://www.mathauditor.com/simpsons-rule-calculator.html Show a screen shot of your Simpson's rule answer. c. Do your two answers match to within round-off error? (yes or no)