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.

## 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. – Calvin Lin Staff · 2 years, 2 months ago

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– Aditya Kumar · 2 years, 2 months ago

Sir if the question is right can you please prove it?Log in to reply

@Calvin Lin @Pi Han Goh – Aditya Kumar · 2 years, 2 months ago

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– Pi Han Goh · 2 years, 2 months ago

How did you get the right hand side of the equation?Log in to reply

– Aditya Kumar · 2 years, 2 months ago

It was a proof questionLog in to reply

– Pi Han Goh · 2 years, 2 months ago

No not right. Consider completing the square for \(ax- x^2\). Then split the integral into two: one from \(0\) to \( \frac a2\) and the other from \(\frac a2\) to \(a\).Log in to reply

– Aditya Kumar · 2 years, 2 months ago

Can you please explain why to split the integral?Log in to reply

– Pi Han Goh · 2 years, 2 months ago

It's easier.Log in to reply

– Aditya Kumar · 2 years, 2 months ago

If u don't mind can u post the solution. I'm getting a integral of \(e^{x^2}\). PleaseLog in to reply

– Pi Han Goh · 2 years, 2 months ago

Yes that's the point, you want to isolate \(e^{x^2} \). Have you got the equation: \( ax-x^2 = -\left(x-\frac a2\right)^2- \frac{a^2}4\)?Log in to reply

– Aditya Kumar · 2 years, 2 months ago

Yes. Can you solve the integral pleaseLog in to reply

– Pi Han Goh · 2 years, 2 months ago

Let \(y = x - \frac a 2\). Change the upper and lower limits. Now, what's the last step?Log in to reply

– Aditya Kumar · 2 years, 2 months ago

But that does not work out. I guess the question is wrongLog in to reply

– Pi Han Goh · 2 years, 2 months ago

What have you tried? Tell me where you got stuck.Log in to reply