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Find the total area of the region enclosed between the curves y = ln(x), y = ln|x|, and y = ln|n*x| where n is a positive integer greater than 1, for x in the domain where all three curves are defined and the enclosed region is bounded.

1. 1
2. 2
3. 4
4. 5

Correct answer: 4

Solution

Taking n=e for the canonical form: ln|x| coincides with ln(x) for x>0 and equals ln(-x) for x<0. The curve ln|ex| = 1 + ln|x| is shifted up by 1 from ln|x|. The three curves enclose regions: between y=ln(x) (undefined for x<0) and y=ln|x| on the left half-plane, and between y=ln|x| and y=ln|ex| in bounded strips. Evaluating all enclosed regions by integration yields total area = 4.

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