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Find an antiderivative of 5x²−5x⁴/x⁵ in the variable x where x>0x>0?

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Final answer:

The antiderivative of the function 5x² - 5x⁴/x⁵ is F(x) = -5/(2x²) - 5 log(x) + C, where C is the integration constant.

Step-by-step explanation:

To find an antiderivative of the function 5x² - 5x⁴/x⁵, let's first simplify the expression. The given function can be simplified to:

5x²/x⁵ - 5x⁴/x⁵ = 5/x³ - 5/x.

Now, we find antiderivatives term by term. For the first term, we use the power rule for integration, and for the second term, we integrate knowing that the antiderivative of 1/x is log(x):

∫ (5/x³)dx = 5 ∫ x⁻³ dx = 5(-1/2)x⁻² = -5/(2x²).

For the second term, ∫ (-5/x)dx = -5 ∫ (1/x)dx = -5 log(x).

Combining both terms gives us the antiderivative:

F(x) = -5/(2x²) - 5 log(x) + C,

where C is the constant of integration.

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