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Let g(x) be an integrable function such that ∫ g(x) dx = g(x). Consider the following statements: Statement 1: ∫ g(x) [f(x)-f''(x)] dx = g(x) [f(x)-f'(x)] + C Statement 2: ∫ g(x) [f(x)+f'(x)] dx = g(x)f(x) + C Which of the following is correct?

1. Statement 1 is true, Statement 2 is true, and Statement 2 correctly explains Statement 1
2. Statement 1 is true, Statement 2 is true, but Statement 2 does not correctly explain Statement 1
3. Statement 1 is false, Statement 2 is true
4. Statement 1 is true, Statement 2 is false

Correct answer: Statement 1 is true, Statement 2 is true, and Statement 2 correctly explains Statement 1

Solution

From int g dx=g we get g'=g. Then d/dx[g f]=g'f+g f'=g(f+f'), proving Statement 2. Writing h=f-f', note h+h'=f-f'', so int g(f-f'')dx=int g(h+h')dx=g h+C=g(f-f')+C, proving Statement 1 directly from Statement 2. Hence both are true and Statement 2 correctly explains Statement 1.

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