Mathematics

$$\int _{ -4 }^{ -5 }{ { e }^{ { \left( x+5 \right)  }^{ 2 } } } dx+\int _{ 1/3 }^{ 2/3 }{ { e }^{ { 9\left( x-\cfrac { 2 }{ 3 }  \right)  }^{ 2 } } } dx$$ is equal to


ANSWER

$$0$$


SOLUTION
We have given from the question
$$\begin{matrix} \int  _{ -4 }^{ -5 }{ { e^{ { { \left( { x+5 } \right)  }^{ 2 } } } }dx }+\int  _{ \frac { 1 }{ 3 }  }^{ \frac { 2 }{ 3 }  }{ { e^{ 9{ { \left( { x-\frac { 2 }{ 3 }  } \right)  }^{ 2 } } } }dx } \\ Let \\ x+5=t \\ Now, \\ \int  _{ 1 }^{ 0 }{ { e^{ { t^{ 2 } } } }dx }\, \, \, +3\int  _{ \frac { 1 }{ 3 }  }^{ \frac { 2 }{ 3 }  }{ { e^{ { { \left( { 3x-2 } \right)  }^{ 2 } } } }dx } \\ Again \\ Let\, \, 3x-2=v \\ \therefore \frac { { dv } }{ { dx } } =3 \\ Now \\ \int  _{ 1 }^{ 0 }{ { e^{ { t^{ 2 } } } }dx }\, \, \, +\frac { 1 }{ 3 } \times 3\int  _{ -1 }^{ 0 }{ { e^{ { v^{ 2 } } } }dv } \\ \int  _{ 1 }^{ 0 }{ { e^{ { t^{ 2 } } } }dx }\, \, \, +\int  _{ -1 }^{ 0 }{ { e^{ { v^{ 2 } } } }dv }\, \, \, \, \, \, \, \, \, \, \, \left[ { u=-x } \right]  \\ \int  _{ 1 }^{ 0 }{ { e^{ { t^{ 2 } } } }dx }\, \, \, +\int  _{ 1 }^{ 0 }{ { e^{ { v^{ 2 } } } }dv }\, \, \, \, =0\, \,  \\  \end{matrix}$$

Hence, the option $$(C)$$ is the correct answer.

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Single Correct Medium Published on 17th 09, 2020
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