# How do you integrate #(6x^5 -2x^4 + 3x^3 + x^2 - x-2)/x^3#?

Thus, our integral becomes

Integrate, recalling that

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To integrate ((6x^5 - 2x^4 + 3x^3 + x^2 - x - 2)/x^3), you can rewrite it as (\frac{6x^5}{x^3} - \frac{2x^4}{x^3} + \frac{3x^3}{x^3} + \frac{x^2}{x^3} - \frac{x}{x^3} - \frac{2}{x^3}). Then, integrate each term separately.

(\int \frac{6x^5}{x^3} dx - \int \frac{2x^4}{x^3} dx + \int \frac{3x^3}{x^3} dx + \int \frac{x^2}{x^3} dx - \int \frac{x}{x^3} dx - \int \frac{2}{x^3} dx)

This simplifies to:

(6 \int x^2 dx - 2 \int x dx + 3 \int dx + \int \frac{1}{x} dx - \int \frac{2}{x^3} dx)

(= 6\frac{x^3}{3} - 2\frac{x^2}{2} + 3x + \ln|x| + \frac{1}{2x^2} + C)

(= 2x^3 - x^2 + 3x + \ln|x| + \frac{1}{2x^2} + C)

(= \frac{2x^3 - x^2 + 3x}{2} + \ln|x| + \frac{1}{2x^2} + C)

Therefore, the integral of ((6x^5 - 2x^4 + 3x^3 + x^2 - x - 2)/x^3) is (\frac{2x^3 - x^2 + 3x}{2} + \ln|x| + \frac{1}{2x^2} + C), where (C) is the constant of integration.

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To integrate the function ( \frac{{6x^5 - 2x^4 + 3x^3 + x^2 - x - 2}}{{x^3}} ), you can perform polynomial division to simplify the expression, then integrate each term separately. After polynomial division, you'll have (6x^2 - 2x + 3 + \frac{1}{x} - \frac{1}{x^2} - \frac{2}{x^3}). Now, you can integrate each term separately:

[ \int 6x^2 ,dx = 2x^3 + C ] [ \int -2x ,dx = -x^2 + C ] [ \int 3 ,dx = 3x + C ] [ \int \frac{1}{x} ,dx = \ln|x| + C ] [ \int \frac{-1}{x^2} ,dx = \frac{1}{x} + C ] [ \int \frac{-2}{x^3} ,dx = -\frac{1}{x^2} + C ]

So, the integral of ( \frac{{6x^5 - 2x^4 + 3x^3 + x^2 - x - 2}}{{x^3}} ) is:

[ 2x^3 - x^2 + 3x + \ln|x| - \frac{1}{x} - \frac{1}{x^2} + C ]

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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.

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.

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.

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