Mathematics

Evaluate: $$\displaystyle \int\limits_3^5 {\frac{{{x^2}}}{{{x^2} - 4}}} dx$$


ANSWER

$$2 - {\log _e}\left( {\frac{{15}}{7}} \right)$$


SOLUTION
$$\frac { { x }^{ 2 } }{ { x }^{ 2 }-4 } =\frac { { x }^{ 2 }-4+4 }{ { x }^{ 2 }-4 } =1+\frac { 4 }{ { x }^{ 2 }-4 } =1+\frac { 4 }{ (x+2)(x-2) }= 1+\frac { 1 }{ (x+2) } -\frac { 1 }{ (x-2) } $$
Thus, $$\int { \frac { { x }^{ 2 } }{ { x }^{ 2 }-4 } dx=\int { \left( 1+\frac { 1 }{ \left( x+2 \right)  } -\frac { 1 }{ \left( x-2 \right)  }  \right) dx }  } $$
$$\int _{ 3 }^{ 5 }{ \frac { { x }^{ 2 } }{ { x }^{ 2 }-4 }  } dx=\left[ x+\log { (x+2) } -\log { (x-2) }  \right] \overset { 5 }{ \underset { 3 }{  }  } $$
$$= \left[ 5+\log { 7 } -\log { 3 }  \right] -\left[ 3+\log { 5 } -\log { 1 }  \right] =2+\log{\dfrac{7}{15}}=2-\log{\dfrac{15}{7}}$$
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Single Correct Medium Published on 17th 09, 2020
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