# How do you find the indefinite integral of ∫ ( x^2 - 9x + 14 )dx / (x^2-4x+3) (x-4) ?

applying the Partial Fraction Method.

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To find the indefinite integral of ( \int \frac{x^2 - 9x + 14}{(x^2 - 4x + 3)(x - 4)} , dx ), you can use partial fraction decomposition. First, factor the denominator into linear factors:

( x^2 - 4x + 3 = (x - 3)(x - 1) )

Now, rewrite the original integral with the partial fraction decomposition:

( \int \frac{x^2 - 9x + 14}{(x - 3)(x - 1)(x - 4)} , dx = \int \left( \frac{A}{x - 3} + \frac{B}{x - 1} + \frac{C}{x - 4} \right) , dx )

Next, find the constants ( A ), ( B ), and ( C ) by multiplying both sides by the denominator and equating coefficients.

Once you have found the values of ( A ), ( B ), and ( C ), integrate each term separately. The integral of ( \frac{A}{x - 3} ) is ( A \ln|x - 3| ), the integral of ( \frac{B}{x - 1} ) is ( B \ln|x - 1| ), and the integral of ( \frac{C}{x - 4} ) is ( C \ln|x - 4| ).

Combine the results to get the final answer.

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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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