Can you evaluate the integral without a calculator?

Question

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∫ (x^(2))/(x^(3)-5) dx

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Answer 1

let u = x³-5, du = 3x²dx

When x=0, u = -5

When x=1, u=-4

∫_-4^-5 (1/3)(1/u)du = (1/3)ln|u|_-4^-5 = (1/3)(ln|-5| -ln|-4|) = (1/3)ln(5/4)

Beyond this you'll need a calculator or a natural log lookup table to find ln(5/4) to finish calculating the answer.

So maybe the answer to the question is "no".

Answer 2

∫(x^(2))/(x^(3)-5) dx

=(1/3)*ln(x³-5) since the numerator is just a constant multiplying term away from being d(x³-5)/dx

Answer 3

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∫ (x^(2))/(x^(3)-5) dx =1/3∫₀¹ (3x^(2))/(x^(3)-5)dx 1/3lnIx³ - 5I₀¹ = 1/3ln4 - 1/3ln5 = 1/3ln0.8 = - 0.0744

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