# How do you integrate #int (2x-2)/(x^2 - 6x +10)^(1/2)dx# using partial fractions?

complete the square for the second

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To integrate (\int \frac{{2x - 2}}{{(x^2 - 6x + 10)^{\frac{1}{2}}}} , dx) using partial fractions, you can first express the integrand as (\frac{{A}}{{(x - 3 + i)(x - 3 - i)}}). Then, solving for A, you find that (A = -\sqrt{2}i).

After that, you can use the substitution method to integrate, where you set (u = x - 3 + i) and (du = dx), leading to the integral (-\sqrt{2}i \int \frac{{du}}{{u}}). This integral is straightforward to solve, resulting in (-\sqrt{2}i \ln|u| + C), where (C) is the constant of integration.

Finally, substitute back (u) and simplify to obtain the final result of (-\sqrt{2}i \ln|x - 3 + i| + C).

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