Isn't the question wrong?

We have to prove that

0aeaxx2dx=ea240aex24dx\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:

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

for RHS.

Therefore I got

ea240aex24dx=0ae2axx24dx\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.

Note by Aditya Kumar
4 years, 4 months ago

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1 vote

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You haven't proven or disproven the statement.

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

Calvin Lin Staff - 4 years, 4 months ago

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

Aditya Kumar - 4 years, 4 months ago

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@Calvin Lin @Pi Han Goh

Aditya Kumar - 4 years, 4 months ago

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

Pi Han Goh - 4 years, 4 months ago

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It was a proof question

Aditya Kumar - 4 years, 4 months ago

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@Aditya Kumar No not right. Consider completing the square for axx2ax- x^2. Then split the integral into two: one from 00 to a2 \frac a2 and the other from a2\frac a2 to aa.

Pi Han Goh - 4 years, 4 months ago

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@Pi Han Goh Can you please explain why to split the integral?

Aditya Kumar - 4 years, 4 months ago

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@Aditya Kumar It's easier.

Pi Han Goh - 4 years, 4 months ago

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@Pi Han Goh If u don't mind can u post the solution. I'm getting a integral of ex2e^{x^2}. Please

Aditya Kumar - 4 years, 4 months ago

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@Aditya Kumar Yes that's the point, you want to isolate ex2e^{x^2} . Have you got the equation: axx2=(xa2)2a24 ax-x^2 = -\left(x-\frac a2\right)^2- \frac{a^2}4?

Pi Han Goh - 4 years, 4 months ago

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@Pi Han Goh Yes. Can you solve the integral please

Aditya Kumar - 4 years, 4 months ago

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@Aditya Kumar Let y=xa2y = x - \frac a 2. Change the upper and lower limits. Now, what's the last step?

Pi Han Goh - 4 years, 4 months ago

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@Pi Han Goh But that does not work out. I guess the question is wrong

Aditya Kumar - 4 years, 4 months ago

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@Aditya Kumar What have you tried? Tell me where you got stuck.

Pi Han Goh - 4 years, 4 months ago

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