Mathematics

Solve:$$\displaystyle \int \dfrac{(x^3+1)(x-2)}{x^2}.dx$$


SOLUTION
Given the integral,
$$\int { \dfrac { ({ x }^{ 3 }+1)(x-2) }{ { x }^{ 2 } }  } dx\\ =\int { \dfrac { { x }^{ 4 }+x-2{ x }^{ 3 }-2 }{ { x }^{ 2 } }  } dx\\ =\int { \left[ { x }^{ 2 }+\dfrac { 1 }{ x } -2x-\dfrac { 2 }{ { x }^{ 2 } }  \right]  } dx\\ =\int { { x }^{ 2 } } dx+\int { \dfrac { 1 }{ x }  } dx-2\int { x } dx-2\int { \dfrac { 1 }{ { x }^{ 2 } }  } dx\\ =\dfrac { { x }^{ 3 } }{ 3 } +\ln { (\left| x \right| ) } -\dfrac { 2{ x }^{ 2 } }{ 2 } -2(-\dfrac { 1 }{ x } )+C\\ =\ln { (\left| x \right| ) } +\dfrac { { x }^{ 3 } }{ 3 } -{ x }^{ 2 }+\dfrac { 2 }{ x } +C\\ =\ln { (\left| x \right| ) } +\dfrac { { x }^{ 3 }-3{ x }^{ 2 } }{ 3 } +\dfrac { 2 }{ x } +C.$$
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Subjective Medium Published on 17th 09, 2020
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