Consider the indefinite integral I= 3x^ 2 (x^ 3 ) ,dx

MATHEMATICS · Subjective · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

Consider the indefinite integral I=∫3x2cos⁡(x3) dx.I=\int 3x^{2}\cos\bigl(x^{3}\bigr)\,dx.Three students suggest the following possible changes of variable for II: • Candidate A: u1=x3u_1=x^{3} • Candidate B: u2=cos⁡(x3)u_2=\cos\bigl(x^{3}\bigr) • Candidate C: u3=xu_3=x a) For each candidate uku_k (k=1,2,3)(k=1,2,3), compute dukdx\dfrac{du_k}{dx} and write the corresponding differential dukdu_k. b) For each uku_k, describe algebraically what the integrand would look like after the change of variable is performed (you do NOT need to carry out the entire antiderivative). c) Determine which candidate leads to the simplest single–step antiderivative once the change of variable is complete. Justify your conclusion by referring to the structural relationship between the original integrand and dukdx\dfrac{du_k}{dx}.

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