(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).ChatGPT response: 1 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 adr = 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) f(1/(1 + u))du = (1/2) In|1 + u|+ C 10 Substituting back u=1+x , we get: 11 (1/2) In |1 + u| + C = (1/2) In|1+1+x?|+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: 1 Method 2: Inverse trig substitution 2 Let's use the substitution x = tan(t). 3 Then, dx = sec2 (t) dt . 4 Our integral becomes: S(x/(1+x2))dx = /(tan(t)/(1 + tan?(t)) sec2(t)dt 5 Simplifying the expression inside the integral using trigonometric identities, we have: 6 S(tan(t)/(1 + tan?(t)) sec2(t)dt = /(tan(t)/ sec2(t) ) dt 7 = f 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+x2) +C. 12 So, the result of the integral using inverse trig substitution is -V(1+x2) +C.ChatGPT response: Method 3: Partial fractions decomposition N This method involves decomposing the rational function into partial fractions. We can rewrite 3 our function as: f(x) = x/(1+ x2) = (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) 6 and equate the numerators: 7 x = A(x2 + 1) + B(x + 1) 8 Expanding and rearranging, we get: *= Ar + A+ Bx + B. 9 Equating the coefficients of like powers of 2, 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 = f(0/(x + 1))da + f(1/(x2 + 1))dx 13 = foda + S(1/(x2 +1))dx 14 = arctan(x) + C. 15 So, the result of the integral using partial fractions decomposition is arctan(x) + C