Mathematics

If $$\displaystyle I_{1}= \int_{-4}^{-5}e^{\left ( x +5 \right )^{2}}dx$$ and $$\displaystyle I_{2}= 3\int_{{1}/{3}}^{{2}/{3}}e^{\left ( 3x -2 \right )^{2}}dx$$ 

then $$I_{1}+I_{2}$$ equals?


ANSWER

$$0$$


SOLUTION
$$I_{ 2 }=3\int _{ \frac { 1 }{ 3 }  }^{ \frac { 2 }{ 3 }  } e^{ \left( 3x-2 \right) ^{ 2 } }dx$$
$$\displaystyle x=\frac { -y-3 }{ 3 } $$ 
$$\displaystyle dx=-\frac { dy }{ 3 } $$ 
$$\displaystyle \Rightarrow I_{ 2 }=3\int _{ \frac { 1 }{ 3 }  }^{ \frac { 2 }{ 3 }  } e^{ \left( 3x-2 \right) ^{ 2 } }dx=-\int _{ -4 }^{ -5 } e^{ \left( y+5 \right) ^{ 2 } }dx=-I_{ 1 }$$ 
$$\displaystyle \Rightarrow I_{ 1 }+I_{ 2 }=0$$
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Single Correct Medium Published on 17th 09, 2020
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