Question

当X＞时,有∫f(x)/xdx=ln(x+√（1+x^2)）+c 求∫xf`(x)dx

Answer 1

∫ f(x)/x dx = ln[x + √(1 + x²)] + C

f(x)/x = d/dx {ln[x + √(1 + x²)] + C} = 1/√(1 + x²)

f(x) = x/√(1 + x²)

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∫ x f'(x) dx

= ∫ x df(x)

= ∫ x d[x/√(1 + x²)]

= ∫ x * 1/(x² + 1)^(3/2) dx

= (1/2)∫ 1/(x² + 1)^(3/2) d(x² + 1)

= (1/2) * (x² + 1)^(-3/2 + 1)/(-3/2 + 1) + C

= -1/√(x² + 1) + C