How do you integrate $\int \frac{x - 5}{{(x - 2)}^{2}}$ $int (x-5)/(x-2)^2$ using partial fractions?

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How do you integrate $\int \frac{x - 5}{{(x - 2)}^{2}}$ $int (x-5)/(x-2)^2$ using partial fractions?

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Answer 1 · 2017-01-24T14:18:48

$\int \frac{x - 5}{{(x - 2)}^{2}} d x = ln | x - 2 | + \frac{3}{x - 2} + C$ $int (x-5)/((x-2)^2) dx = ln abs (x-2) +3 /(x-2) + C$

Explanation:

We do not really need partial fractions here as we can separate the numerator by writing:

$\int \frac{x - 5}{{(x - 2)}^{2}} d x = \int \frac{x - 2 - 3}{{(x - 2)}^{2}} d x = \int \frac{d x}{x - 2} - 3 \int \frac{d x}{{(x - 2)}^{2}}$ $int (x-5)/((x-2)^2) dx = int (x-2-3)/((x-2)^2) dx = int (dx)/(x-2) -3 int (dx)/((x-2)^2)$

now, as $d (x - 2) = d x$ $d(x-2) = dx$ we can solve the integrals directly

$\int \frac{x - 5}{{(x - 2)}^{2}} d x = ln | x - 2 | + \frac{3}{x - 2} + C$ $int (x-5)/((x-2)^2) dx = ln abs (x-2) +3 /(x-2) + C$

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How do you integrate $\int \frac{x - 5}{{(x - 2)}^{2}}$ $int (x-5)/(x-2)^2$ using partial fractions?

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How do you integrate $\int \frac{x - 5}{{(x - 2)}^{2}}$ $int (x-5)/(x-2)^2$ using partial fractions?