Mathematics

Problem

For each integral listed below, a student must plan the evaluation before beginning any calculations. (i) For each integral (a)–(c), state the technique you would apply first and give one clear sentence to justify that choice. (a)

\displaystyle \int x\,\ln\!(x^{2}+1)\,dx

(b)

\displaystyle \int \frac{x^{2}+2x+3}{x+1}\,dx

(c)

\displaystyle \int x\,e^{x^{2}}\sin(x^{2})\,dx

(ii) For integral (a) only, explain in two or three sentences why beginning with any one different standard technique would lead to a longer or more complicated solution path. a. (i) Identify the most suitable starting technique for each integral (a)–(c) and justify your selection in one sentence for each. b. (ii) Focusing on integral (a), explain briefly why an alternative first step would make the evaluation less efficient.

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