Mathematics

Evaluate the integrals using substitution.
$$\displaystyle\int^2_0x\sqrt{x+2}$$ (Put $$x+2=t^2$$).


SOLUTION
$$\int _{ 0 }^{ 2 }{ x\sqrt { x+2 }  } dx$$
Let $$x+2={ t }^{ 2 }\Rightarrow dx=2tdt\quad $$
$$=\int _{ \sqrt { 2 }  }^{ 2 }{ \left( { t }^{ 2 }-2 \right)  } .t.2t.dt=2\int _{ \sqrt { 2 }  }^{ 2 }{ \left( { t }^{ 4 }-2{ t }^{ 2 } \right)  } dt=2{ \left[ \cfrac { { t }^{ 5 } }{ 5 } -\cfrac { 2{ t }^{ 3 } }{ 3 }  \right]  }_{ \sqrt { 2 }  }^{ 2 }$$
$$=2\left[ \left( \cfrac { 32 }{ 5 } -\cfrac { 2\times 8 }{ 3 }  \right) -\left( \cfrac { 4\sqrt { 2 }  }{ 5 } -\cfrac { 2.2\sqrt { 2 }  }{ 3 }  \right)  \right] =2\left( \cfrac { 16 }{ 15 } +\cfrac { 8\sqrt { 2 }  }{ 15 }  \right) =\cfrac { 32+16\sqrt { 2 }  }{ 15 } $$
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Subjective Medium Published on 17th 09, 2020
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