Mathematics

Evaluate:
$$\displaystyle\int\dfrac{2x+3}{\sqrt{4x+3}}dx=$$


SOLUTION
$$\int { \dfrac { 2x+3 }{ \sqrt { 4x+3 }  }  } dx$$
$$\Rightarrow \dfrac { 1 }{ 2 } \int { \dfrac { \left( 4x+3 \right)  }{ \sqrt { 4x+3 }  }  } +\dfrac { 3 }{ 2 } \int { \dfrac { dx }{ \sqrt { 4x+3 }  }  } $$
$$\Rightarrow 4x+3=m$$
$$dx=\dfrac { dm }{ 4 } $$
$$\Rightarrow \dfrac { 1 }{ 8 } \int { \dfrac { m }{ \sqrt { m }  } dm } +\dfrac { 3 }{ 8 } \int { \dfrac { dm }{ \sqrt { m }  }  } $$
$$\Rightarrow \dfrac { 1 }{ 8 } \int { \sqrt { m }  } dm+\dfrac { 3 }{ 8 } \int { { \left( \sqrt { m }  \right)  }^{ -1 } } dm$$
$$\Rightarrow \dfrac { 1 }{ 8 } \times \dfrac { 2 }{ 3 } \times { m }^{ 3/2 }+\dfrac { 3 }{ 8 } \times 2\times \sqrt { m } +c$$
$$\Rightarrow \dfrac { 1 }{ 12 } { \left( 4x+3 \right)  }^{ 3/2 }+\dfrac { 3 }{ 4 } \sqrt { 4x+3 } +c$$
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Subjective Hard Published on 17th 09, 2020
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