How to integrate this? #intcolor(white).(3x^2-x+1)/(x^3-x^2)#

Answer 1

Scarlett Allison

#3lnabs(x-1)+1/x+C#

#int(3x^2-x+1)/(x^3-x^2)dx=int(3x^2)/(x^2(x-1))dx -int(x-1)/(x^2(x-1))dx=int(3)/((x-1))dx-int1/x^2dx=3lnabs(x-1)+1/x+C#

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Answer 2

Elijah Abram

To integrate ( \frac{3x^2 - x + 1}{x^3 - x^2} ), follow these steps:

Rewrite the fraction as partial fractions if possible.
Factor the denominator to see if any common factors can be canceled.
If partial fractions are required, set up the partial fraction decomposition and solve for the unknown constants.
Integrate each term separately.
If there are no partial fractions, directly integrate the function.

Since the process can be lengthy and involves several steps, I will provide the result of integrating the given expression:

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

Where ( C ) is the constant of integration.

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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.