Mathematics

Evaluate $$\int x^{2}\log xdx$$


SOLUTION
$$\begin{array}{l} We\, \, have \\ I=\int { { x^{ 2 } }\log x }  \\ =\log  x.\dfrac { { { x^{ 3 } } } }{ 3 } -\int { \dfrac { { { x^{ 3 } } } }{ 3 } .\dfrac { 1 }{ x } dx } \, \, \, \, \, \left[ { \therefore \int { u.vdx=vu-\int { vdu }  }  } \right]  \\ =\dfrac { { { x^{ 3 } } } }{ 3 } \log  x-\dfrac { 1 }{ 3 } \int { { x^{ 2 } }dx }  \\ =\dfrac { { { x^{ 3 } } } }{ 3 } \log  x-\dfrac { 1 }{ 3 } \times \dfrac { { { x^{ 3 } } } }{ 3 }  \\ =\dfrac { { { x^{ 3 } } } }{ 3 } .\log  x-\dfrac { { { x^{ 3 } } } }{ 9 } +C \\ It\, is\, the\, required\, answer. \end{array}$$
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Subjective Medium Published on 17th 09, 2020
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