2026 ASRJC P1 Q5

2026 ASRJC P1 Q5

Junior College 2
7 marks
  1. Find the exact value of \(\displaystyle \int_{-1}^{2}|e^{-x}-x-1|\,\mathrm{d}x\). Show your workings clearly.[3]
    1. Show that \(\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}[\ln(\sec x+\tan x)]=\sec x\) for \(0<x<\frac\pi2\).[1]
    2. Hence, using integration, show that \(\displaystyle \int\sec^3x\,\mathrm{d}x=\frac12\{\sec x\tan x+\ln(\sec x+\tan x)\}+C\) for \(0<x<\frac\pi2\).[3]

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Answer:(a) \(e+e^{-2}+\frac32\) (b)(i) derivative \(=\sec x\) (b)(ii) \(\int\sec^3x\,\mathrm{d}x=\frac12\{\sec x\tan x+\ln(\sec x+\tan x)\}+C\)

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