Waste less time on Facebook — follow Brilliant.
×

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.

Note by Aditya Kumar
1 year, 7 months ago

No vote yet
1 vote

Comments

Sort by:

Top Newest

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 · 1 year, 7 months ago

Log in to reply

@Calvin Lin Sir if the question is right can you please prove it? Aditya Kumar · 1 year, 7 months ago

Log in to reply

@Calvin Lin @Pi Han Goh Aditya Kumar · 1 year, 7 months ago

Log in to reply

@Aditya Kumar How did you get the right hand side of the equation? Pi Han Goh · 1 year, 7 months ago

Log in to reply

@Pi Han Goh It was a proof question Aditya Kumar · 1 year, 7 months ago

Log in to reply

@Aditya Kumar 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 \(a\). Pi Han Goh · 1 year, 7 months ago

Log in to reply

@Pi Han Goh Can you please explain why to split the integral? Aditya Kumar · 1 year, 7 months ago

Log in to reply

@Aditya Kumar It's easier. Pi Han Goh · 1 year, 7 months ago

Log in to reply

@Pi Han Goh If u don't mind can u post the solution. I'm getting a integral of \(e^{x^2}\). Please Aditya Kumar · 1 year, 7 months ago

Log in to reply

@Aditya Kumar Yes that's the point, you want to isolate \(e^{x^2} \). Have you got the equation: \( ax-x^2 = -\left(x-\frac a2\right)^2- \frac{a^2}4\)? Pi Han Goh · 1 year, 7 months ago

Log in to reply

@Pi Han Goh Yes. Can you solve the integral please Aditya Kumar · 1 year, 7 months ago

Log in to reply

@Aditya Kumar Let \(y = x - \frac a 2\). Change the upper and lower limits. Now, what's the last step? Pi Han Goh · 1 year, 7 months ago

Log in to reply

@Pi Han Goh But that does not work out. I guess the question is wrong Aditya Kumar · 1 year, 7 months ago

Log in to reply

@Aditya Kumar What have you tried? Tell me where you got stuck. Pi Han Goh · 1 year, 7 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...