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The integral ∫ (1 + x - 1/x) e^(x + 1/x) dx is equal to

1. (1) -x e^(x + 1/x) + c
2. (2) (x - 1) e^(x + 1/x) + c
3. (3) x e^(x + 1/x) + c
4. (4) (x + 1) e^(x + 1/x) + c

Correct answer: (3) x e^(x + 1/x) + c

Solution

The correct option is right because when differentiating the expression x e^(x + 1/x), the product rule and the chain rule yield the integrand (1 + x - 1/x) e^(x + 1/x), confirming that it is indeed the antiderivative.

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