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.