Mathematics

Evaluate : 
$$I = \int {\dfrac{{2x}}{{{x^2} - 6x + 6}}dx} $$


SOLUTION
$$\int \dfrac  { 2 x } { x ^ { 2 } - 6 x + 6 } d x$$

put $$x ^ { 2 } - 6 x + 6 = t$$

$$( 2 x - 6 ) d x = d t$$
$$\int\dfrac  { 2 x } { x ^ { 2 } - 6 x + 6 } d x$$
$$= \int \dfrac  { 2 x - 6 } { x ^ { 2 } - 6 x + 6 } d x + \int \dfrac  { 6 } { x ^ { 2 } - 6 x + 6 } d x$$

$$= \int \dfrac  { d t } { t } + \int \dfrac  { 6 } { ( x - 3 ) ^ { 2 } - ( \sqrt { 3 } ) ^ { 2 } } d x$$

$$= \log \left( x ^ { 2 } - 6 x + 6 \right) + 6 \cdot \dfrac  { 1 } { 2 \cdot \sqrt { 3 } } \log \left( \dfrac  { x - 3 - \sqrt { 3 } } { x - 3 + \sqrt { 3 } } \right) + c$$

$$= \log \left( x ^ { 2 } - 6 x + 6 \right) + \sqrt { 3 } \log \left( \dfrac  { x - 3 -\sqrt { 3 } } { x - 3 + \sqrt { 3 } } \right) + c$$
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Subjective Medium Published on 17th 09, 2020
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