Mathematics

Evaluate the following:
$$\displaystyle \int (3x + 1)\sqrt{2x - 1}dx$$


SOLUTION

$$I=\int { \left( 3x+1 \right) \sqrt { 2x-1 }  } dx\\ Put\quad 2x-1=t\\ 2dx=dt\\ dx=\cfrac { 1 }{ 2 } dt\\ \therefore 3x+1=\left( t+1 \right) \cfrac { 3 }{ 2 } +1\\ \therefore I=\cfrac { 1 }{ 2 } \int { \left( \cfrac { 3 }{ 2 } t+\cfrac { 3 }{ 2 } +1 \right) \sqrt { t } dt } \\ I=\cfrac { 1 }{ 2 } \int { \left( \cfrac { 3 }{ 2 } { t }^{ \cfrac { 3 }{ 2 }  }+\cfrac { 5 }{ 2 } { t }^{ \cfrac { 1 }{ 2 }  } \right) dt } \\ I=\cfrac { 1 }{ 2 } \left( \cfrac { 3 }{ 2 } \times \cfrac { 2 }{ 5 } { t }^{ \cfrac { 5 }{ 2 }  }+\cfrac { 5 }{ 2 } \times \cfrac { 2 }{ 3 } { t }^{ \cfrac { 3 }{ 2 }  } \right) +C\quad \left( \because \int { { x }^{ n }dx=\cfrac { 1.{ x }^{ n+1 } }{ n+1 }  }  \right) \\ I=\cfrac { 3 }{ 10 } { t }^{ \cfrac { 5 }{ 2 }  }+\cfrac { 5 }{ 6 } { t }^{ \cfrac { 3 }{ 2 }  }+C\\ I=\cfrac { 3 }{ 10 } { \left( 2x-1 \right)  }^{ ^{ \cfrac { 5 }{ 2 }  } }+\cfrac { 5 }{ 6 } { \left( 2x-1 \right)  }^{ \cfrac { 3 }{ 2 }  }+C$$

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Subjective Medium Published on 17th 09, 2020
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