Mathematics

Integrate:
$$\frac { 3 x + 5 } { ( x + 1 ) ( x - 2 ) ^ { 2 } } d x$$


SOLUTION
Let  $$\dfrac { 3x+5 }{ \left( x+1 \right) { \left( x-2 \right)  }^{ 2 } } =\dfrac { A }{ x+1 } +\dfrac { B }{ x-2 } +\dfrac { C }{ { \left( x-2 \right)  }^{ 2 } } $$
On comparing, we get
$$A=\dfrac { 2 }{ 9 } $$,  $$B=\dfrac { -2 }{ 9 } $$,  $$C=\dfrac { 11 }{ 3 } $$
So,  $$\int { \dfrac { \left( 3x+5 \right)  }{ \left( x+1 \right) { \left( x-2 \right)  }^{ 2 } } dx } =\int { \dfrac { 2 }{ 9\left( x+1 \right)  } dx } +\int { \dfrac { -2 }{ 9\left( x-2 \right)  } dx } +\int { \dfrac { 11 }{ 3{ \left( x-2 \right)  }^{ 2 } } dx } $$
So,  $$\int { \dfrac { 3x+5 }{ \left( x+1 \right) { \left( x-2 \right)  }^{ 2 } }  } =\dfrac { 2 }{ 9 } log\left| x+1 \right| -\dfrac { 2 }{ 9 } ln\left( x-2 \right) -\dfrac { 11 }{ 3 } \dfrac { 1 }{ \left( x-2 \right)  } +C$$
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Subjective Medium Published on 17th 09, 2020
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