We have to prove that

\(\displaystyle \int _{ 0 }^{ a }{ { e }^{ ax-{ x }^{ 2 } } } dx\quad ={ e }^{ \frac { { a }^{ 2 } }{ 4 } }\int _{ 0 }^{ a }{ { e }^{ \frac { -{ x }^{ 2 } }{ 4 } } } dx\)

I used the property:

\(\displaystyle \int _{ 0 }^{ a }{ f(x) } dx\quad =\quad \int _{ 0 }^{ a }{ f(a-x) } dx\)

for RHS.

Therefore I got

\(\displaystyle { e }^{ \frac { { a }^{ 2 } }{ 4 } }\int _{ 0 }^{ a }{ { e }^{ \frac { -{ x }^{ 2 } }{ 4 } } } dx\quad =\quad \int _{ 0 }^{ a }{ { e }^{ \frac { 2ax-{ x }^{ 2 } }{ 4 } } } dx\) Please correct me if I'm wrong. If I'm right please do mention it.

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## Comments

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TopNewestYou haven't proven or disproven the statement.

E.g. If you are asked to show that \( 2 + 2 = 2 \times 2 \), then saying that \( 2 + 2 = 4 \) doesn't mean that the initial statement must be wrong.

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Sir if the question is right can you please prove it?

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@Calvin Lin @Pi Han Goh

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How did you get the right hand side of the equation?

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It was a proof question

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