# Chapter 4 - Applications of the Derivative - 4.9 Antiderivatives - 4.9 Exercises: 35

$${x^4} - 3{x^2} + C$$

#### Work Step by Step

\eqalign{ & \int {\frac{{4{x^4} - 6{x^2}}}{x}dx} \cr & {\text{split the numerator}} \cr & = \int {\left( {\frac{{4{x^4}}}{x} - \frac{{6{x^2}}}{x}} \right)dx} \cr & = \int {\left( {4{x^3} - 6x} \right)dx} \cr & {\text{use power rule for indefinite integrals}} \cr & = 4\left( {\frac{{{x^4}}}{4}} \right) - 6\left( {\frac{{{x^2}}}{2}} \right) + C \cr & {\text{simplify}} \cr & = {x^4} - 3{x^2} + C \cr & {\text{check by differentiation}} \cr & {\text{ = }}\frac{d}{{dx}}\left( {{x^4} - 3{x^2} + C} \right) \cr & = 4{x^3} - 6x \cr}

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