How do you integrate #int (2x-2)/ (x^2 - 4x + 13)dx# using partial fractions?
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To integrate (\int \frac{2x - 2}{x^2 - 4x + 13} , dx) using partial fractions, follow these steps:
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Factor the denominator: (x^2 - 4x + 13 = (x - 2)^2 + 9).
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Rewrite the integrand using partial fractions: [\frac{2x - 2}{x^2 - 4x + 13} = \frac{A(x - 2) + B}{(x - 2)^2 + 9}]
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Multiply both sides by the denominator ((x^2 - 4x + 13)) to clear the fraction.
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Simplify the resulting equation and solve for (A) and (B).
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Once you have (A) and (B), rewrite the original integral in terms of the partial fractions:
[\int \frac{2x - 2}{x^2 - 4x + 13} , dx = \int \frac{A(x - 2) + B}{(x - 2)^2 + 9} , dx]
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Use trigonometric substitution or another appropriate method to integrate the expression involving (A) and (B).
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Finally, integrate the resulting expression.
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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.
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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