Question 2. In this question you will continue to explore using multiple techniques to solve an integral, while also critically examining the output of ChatGPT's _ solutions to the following prompt. Prompt. Integrate x/(1 + x2) using three different methods: first u-substitution, second inverse trig substitution, and then third using a partial fractions decomposition. (1) Warmup. What differences, if any, are there between the integral from Question 1 and the integral for Question 2? Do you think you will be able to use the same methods to solve both integrals? (2) Are the final results of the three ChatGPT solutions 2 (on the next 3 pages) equivalent? What can you conclude from this? (3) For each of the three ChatGPT solutions either confirm that the solution is correct, or identify the first error in the solution (by citing the line number where the error appears and explaining what the error is). (4) For all of the incorrect solutions, correct any errors, and then (if possible) write up a corrected version of ChatGPT's solution. (For the purpose of this exercise, stick as close as you can to the original solution only making minimal, necessary changes.) (5) If any solution is impossible to make into a correct solution using minimal corrections, then briefly explain why.ChatGPT response: Sure, here are the three different methods to integrate the function f(x) =x/(1+2). 2 Method 1: u-substitution 3 Let's use the substitution u =1+x2. 4 Then, du = 2xdx. 5 Rearranging, we have xde = du/2. Now we can write our integral as: 6 S(x/(1 + x2))dx = S(1/(1 + u))(du/2) 7 = (1/2) S(1/(1 + u))du 8 Now, we can integrate this new expression with respect to u: 9 (1/2) S(1/(1 + u))du = (1/2) In |1 + ul + C 10 Substituting back u =1+x2, we get : 11 (1/2) In |1 + u| + C = (1/2) In|1+1+x2| +C 12 = (1/2) In |2+ x2 + C 13 So, the result of the integral using u-substitution is (1/2) In 2+x?| +C.ChatGPT response: Method 2: Inverse trig substitution 2 Let's use the substitution x = tan(t). 3 Then, da = sec2 (t) dt. becomes : S(x/(1 + x2))dx = S(tan(t)/(1 + tan2(t)) sec2(t) dt 5 Simplifying the expression inside the integral using trigonometric identities, we have: 6 (tan(t)/(1 + tan2(t)) sec2(t)dt = S(tan(t)/ sec2(t))dt 7 = [ sin(t) dt 8 = - cos(t) + C 9 Substituting back x = tan(t), we get: 10 - cos(t) + C = - cos(arctan(x)) + C 11 = -V ( 1 + 12 ) + C . 12 So, the result of the integral using inverse trig substitution is -V(1+2) +C.ChatGPT response: Method 3: Partial fractions decomposition 2 This method involves decomposing the rational function into partial fractions. We can rewrit 3 our function as: 4 f (2) = 2/(1+ 22) = (A/(x+1)) +(B/(x2 +1)) 5 To find the values of A and B, we can multiply both sides by the denominator (x +1)(x2 +1) and equate the numerators: 7 x = A(x2 + 1) + B(x +1) 8 Expanding and rearranging, we get: x = Ar' + A+ Bx + B. 9 Equating the coefficients of like powers of x, we have: A=0, A+B=1. 10 From which, we find that A =0 and B =1. 11 Now, we can rewrite our integral as the sum of the integrals of the partial fractions: 12 S(x/(1+ x2))dx = S(0/(x + 1))da + S(1/(x2 + 1))dx 13 = fodx + S(1/(x2 + 1))dx 14 = arctan(x) + C. 15 So, the result of the integral using partial fractions decomposition is arctan(x) + C