Evaluate: integral of (1)/(x^2-7x+10) x

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Expression: $\int{ \frac{ 1 }{ {x}^{2}-7x+10 } } \mathrm{d} x$

Rewrite the fraction using partial-fraction decomposition

$\int{ -\frac{ 1 }{ 3\left( x-2 \right) }+\frac{ 1 }{ 3\left( x-5 \right) } } \mathrm{d} x$

Use the property of integral $\int{ f\left( x \right)\pmg\left( x \right) } \mathrm{d} x=\int{ f\left( x \right) } \mathrm{d} x\pm\int{ g\left( x \right) } \mathrm{d} x$

$-\int{ \frac{ 1 }{ 3\left( x-2 \right) } } \mathrm{d} x+\int{ \frac{ 1 }{ 3\left( x-5 \right) } } \mathrm{d} x$

Evaluate the indefinite integral

$-\frac{ 1 }{ 3 } \times \ln\left({|x-2|}\right)+\int{ \frac{ 1 }{ 3\left( x-5 \right) } } \mathrm{d} x$

Evaluate the indefinite integral

$-\frac{ 1 }{ 3 } \times \ln\left({|x-2|}\right)+\frac{ 1 }{ 3 } \times \ln\left({|x-5|}\right)$

Add the constant of integration $C \in ℝ$

$\begin{array} { l }-\frac{ 1 }{ 3 } \times \ln\left({|x-2|}\right)+\frac{ 1 }{ 3 } \times \ln\left({|x-5|}\right)+C,& C \in ℝ\end{array}$

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