# How do you integrate #int (x-3x^2)/((x-6)(x-2)(x+4)) # using partial fractions?

The answer is

Let's perform the decomposition into partial fractions

Therefore,

So,

Therefore,

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To integrate ( \int \frac{x - 3x^2}{(x - 6)(x - 2)(x + 4)} ) using partial fractions, we follow these steps:

- Perform long division if necessary to ensure the degree of the numerator is less than the degree of the denominator.
- Factor the denominator into linear and irreducible quadratic factors.
- Express the given fraction as a sum of simpler fractions using partial fraction decomposition.
- Find the unknown constants by equating coefficients.
- Integrate each partial fraction term separately.

Given the integral ( \int \frac{x - 3x^2}{(x - 6)(x - 2)(x + 4)} ), let's start by performing partial fraction decomposition:

[ \frac{x - 3x^2}{(x - 6)(x - 2)(x + 4)} = \frac{A}{x - 6} + \frac{B}{x - 2} + \frac{C}{x + 4} ]

To find the constants ( A ), ( B ), and ( C ), we multiply both sides by the common denominator ( (x - 6)(x - 2)(x + 4) ) and simplify:

[ x - 3x^2 = A(x - 2)(x + 4) + B(x - 6)(x + 4) + C(x - 6)(x - 2) ]

Next, we equate coefficients for like terms:

For ( x^2 ): [ -3 = A + B + C ]

For ( x ): [ 1 = -2A - 6B - 6C ]

For constants: [ 0 = 8A + 24B - 24C ]

Solving this system of equations will give us the values of ( A ), ( B ), and ( C ).

Once we have the values of ( A ), ( B ), and ( C ), we substitute them back into the partial fraction decomposition:

[ \int \frac{x - 3x^2}{(x - 6)(x - 2)(x + 4)} = \int \frac{A}{x - 6} + \frac{B}{x - 2} + \frac{C}{x + 4} ]

Finally, we integrate each partial fraction term separately.

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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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