(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.)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