Mathematics

Mathematics

Problem

A student submits the following solution for a trigonometric integral: ∫0π/4sec⁡2θ dθ=(ln⁡∣tan⁡θ+sec⁡θ∣)∣0π/4=ln⁡(2).\int_{0}^{\pi/4} \sec^{2}\theta\,d\theta = \left( \ln|\tan\theta + \sec\theta| \right)\Big|_{0}^{\pi/4} = \ln(\sqrt{2}). You are asked to review the work. a. Differentiate the student’s proposed antiderivative A(θ)=ln⁡∣tan⁡θ+sec⁡θ∣A(\theta)=\ln|\tan\theta+\sec\theta|. Simplify A′(θ)A'(\theta) and determine whether it equals sec⁡2θ\sec^{2}\theta on the open interval (0,π/4)(0,\pi/4). b. Based on your calculation in part (a), decide whether the student’s numerical answer ln⁡(2)\ln(\sqrt{2}) is correct. Justify your conclusion carefully. c. Evaluate ∫0π/4sec⁡2θ dθ\displaystyle \int_{0}^{\pi/4} \sec^{2}\theta\,d\theta correctly and state its exact value. d. Identify and describe the specific error (algebraic, trigonometric, or calculus-related) that led to the student’s incorrect result.

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