Mathematics

$$\int { x+2y } \sqrt { { x }^{ 2 }+5x+6 } $$


SOLUTION
$$\displaystyle \int x+2y \sqrt {x^2 +5x+6}dx$$
$$\Rightarrow \ \displaystyle \int x\ dx +2y \displaystyle \int \sqrt {x^2+5x+6}dx$$
$$\Rightarrow \ \dfrac {x^2}{2}+2y \displaystyle \int \sqrt {\left (x+\dfrac {5}{2}\right)^2 -\dfrac {1}{4}}dx\quad \mu =x+\dfrac {5}{2}$$
$$\Rightarrow \ 2y \dfrac {1}{2} \displaystyle \int \sqrt {4\mu^2 -1}du \quad v= \sec^{-1}(2\mu)$$
$$\Rightarrow \ 2y.\dfrac {1}{2}.\dfrac {1}{2} \displaystyle \int (-1+\sec^2 (v))\sec v\ dv$$
$$\displaystyle \int \sec v \ dv +\displaystyle \int \sec^3 x\ dv$$
$$\dfrac {\sec^2 v \sin v}{2}+\dfrac {1}{2} \displaystyle \int \sec v\ dv$$
$$\Rightarrow \ \dfrac {x^2}{2}\dfrac {y}{2} (-\ln |\tan v+ \sec v|+ \dfrac {\sec^2 v.\sin v}{2}+\dfrac {1}{2} \ln |\tan v+\sec v|)$$

$$\Rightarrow \dfrac { 1 }{ 2 } \left[ { x }^{ 2 }+y\left( \ln { \left| \tan { \left( \sec ^{ -1 }{ \left( 2\left( x+\dfrac { 5 }{ 2 }  \right)  \right)  }  \right)  }  \right|  }  \right) +\sec { \left( \sec ^{ -1 }{ \left( 2\left( x+\dfrac { 5 }{ 2 }  \right)  \right)  }  \right)  }  \right] \left( -1+\dfrac { 1 }{ 2 }  \right) +\sec ^{ 2 }{ \left( \sec ^{ -1 }{ \left( 2\left( x+\dfrac { 5 }{ 2 }  \right)  \right)  }  \right)  } \sin { \left( \sec ^{ -1 }{ 2\left( x+\dfrac { 5 }{ 2 }  \right)  }  \right)  } \\ $$
$$\Rightarrow \dfrac { { x }^{ 2 } }{ 2 } +\dfrac { y }{ 4 } \left[ 2\left( 2x+5 \right) \sqrt { { x }^{ 2 }+5x+6 } -\ln { \left| 2\sqrt { { x }^{ 2 }+5x+6 } +2x+5 \right|  }  \right] +C$$

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Subjective Medium Published on 17th 09, 2020
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