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
2 years, 5 months ago

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 \( 2 \times 3 \)
2^{34} \( 2^{34} \)
a_{i-1} \( a_{i-1} \)
\frac{2}{3} \( \frac{2}{3} \)
\sqrt{2} \( \sqrt{2} \)
\sum_{i=1}^3 \( \sum_{i=1}^3 \)
\sin \theta \( \sin \theta \)
\boxed{123} \( \boxed{123} \)

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 - 2 years, 5 months ago

Log in to reply

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

Aditya Kumar - 2 years, 5 months ago

Log in to reply

@Calvin Lin @Pi Han Goh

Aditya Kumar - 2 years, 5 months ago

Log in to reply

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

Pi Han Goh - 2 years, 5 months ago

Log in to reply

It was a proof question

Aditya Kumar - 2 years, 5 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 - 2 years, 5 months ago

Log in to reply

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

Aditya Kumar - 2 years, 5 months ago

Log in to reply

@Aditya Kumar It's easier.

Pi Han Goh - 2 years, 5 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 - 2 years, 5 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 - 2 years, 5 months ago

Log in to reply

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

Aditya Kumar - 2 years, 5 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 - 2 years, 5 months ago

Log in to reply

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

Aditya Kumar - 2 years, 5 months ago

Log in to reply

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

Pi Han Goh - 2 years, 5 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...