Isn't the question wrong?

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.

#Calculus #Aditya.kumar23

Easy Math Editor

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Comments

You 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 - 5 years, 12 months ago

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