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How do you integrate #f(x)=(4x^3-7x)/(5x^2+2)# using the quotient rule?

Answer 1

Samantha Adkison

There is no quotient rule for integrals.

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Answer 2

Christopher Agresta

To integrate the function ( f(x) = \frac{{4x^3 - 7x}}{{5x^2 + 2}} ) using the quotient rule, follow these steps:

Differentiate the numerator and denominator separately.
Apply the quotient rule, which states: [ \int \frac{{f'(x)}}{{g(x)}} , dx = \ln|g(x)| + C ] where ( f'(x) ) is the derivative of the numerator and ( g(x) ) is the denominator.
Calculate the derivatives of the numerator and denominator: [ f'(x) = \frac{{d}}{{dx}}(4x^3 - 7x) ] [ g'(x) = \frac{{d}}{{dx}}(5x^2 + 2) ]
Use the quotient rule: [ \int \frac{{f'(x)}}{{g(x)}} , dx = \ln|5x^2 + 2| + C ]

Therefore, the integral of ( f(x) ) using the quotient rule is: [ \int \frac{{4x^3 - 7x}}{{5x^2 + 2}} , dx = \ln|5x^2 + 2| + C ]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.