anonymous
  • anonymous
How to find this derivative of this integral?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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anonymous
  • anonymous
\[\int\limits_{-1}^{2} 2xdx\]
anonymous
  • anonymous
\[ \frac{d}{dx}\int^{g(x)}_af(t)dt = f(g(x))g'(x) \]
anonymous
  • anonymous
Do you want the antiderivative?

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anonymous
  • anonymous
I meant how to evaluate it, i read it wrong
anonymous
  • anonymous
Okay first you need to find the anti-derivative. Do you know how to do that?
anonymous
  • anonymous
How?
anonymous
  • anonymous
First, do you know how to find the derivative of the following function: \[ \frac{d}{dx} (x^n) \]
anonymous
  • anonymous
would you just bring the n down?
anonymous
  • anonymous
Yeah, what is the derivative in terms of \(n\)?
anonymous
  • anonymous
it would be one less than n?
anonymous
  • anonymous
One less than \(n\)? Are you saying \(n-1\)? What is the derivative of \(x^n\)?
anonymous
  • anonymous
it would just be n?
anonymous
  • anonymous
Nope. Okay how about this... What is the derivative of \(x^5\)?
anonymous
  • anonymous
\[5x ^{4}\]
anonymous
  • anonymous
Okay, so suppose \(5=n\). What is the derivative of \(x^n\)?
anonymous
  • anonymous
nx? I dont know if i'm understanding?
anonymous
  • anonymous
\[ x^5 \to 5x \]What happened to the \(4\)?
anonymous
  • anonymous
What happens to the exponent?
anonymous
  • anonymous
I'm trying to get you to understand the general concept, then you'll be able to do many problems on your own.
anonymous
  • anonymous
What happens to the 4? isn't it supposed to be the new exponent?
anonymous
  • anonymous
Yes. So what is the new exponent when you take the derivative of \(x^n\)?
anonymous
  • anonymous
n-1?
anonymous
  • anonymous
\[ \frac{d}{dx}x^n = nx^{n-1} \]
anonymous
  • anonymous
Yes, I understand that
anonymous
  • anonymous
Okay, now suppose we want to reverse it...
anonymous
  • anonymous
what do we do?
anonymous
  • anonymous
So suppose we want to find the anti derivative of \(x^m\)
anonymous
  • anonymous
okay
anonymous
  • anonymous
We need to find \(a\) and \(b\):\[ \frac{d}{dx} ax^b = x^m \]
anonymous
  • anonymous
We know that \[ \frac{d}{dx}ax^b = abx^{b-1} \]So \[ \color{red}{ab}x^{\color{blue}{b-1}} = x^{\color{blue}{m}} \]This means two things: \[ b-1 = m\\ ab = 1 \]
anonymous
  • anonymous
We can see that \[ b = m+1 \]So we have found \(b\). But what about \(a\)?\[ a = \frac 1 b = \frac 1 {m+1} \]
anonymous
  • anonymous
So \[ \frac{d}{dx}ax^b = \frac{d}{dx} \frac{1}{m+1} x^{m+1} = x^m \]
anonymous
  • anonymous
In sort, the antiderivative of \(x^m\) is \[ \frac{x^{m+1}}{m+1} \]
anonymous
  • anonymous
okay, so how do i evaluate the integral....?
anonymous
  • anonymous
What is the antiderivative of \(x^1\)?
anonymous
  • anonymous
x^2/2?
anonymous
  • anonymous
Yes. What is the anti derivative of \(2x\) then?
anonymous
  • anonymous
is it x^2?
anonymous
  • anonymous
Yes.
anonymous
  • anonymous
Technically you need a constant of integration so it is \(x^2+C\) Anyway\[ \int^b_af'(x)dx = f(b)-f(a) \]In this case \(f'(x)=2x\) and we have found that \(f(x)=x^2+C\).
anonymous
  • anonymous
The \(C\) will cancel out when we subtract, so we don't need to worry about it.
anonymous
  • anonymous
So what is \(b\) and what is \(a\) in this case?
anonymous
  • anonymous
a is -1 and b is 2
anonymous
  • anonymous
Okay so what is \(f(b)-f(a)\) in this case?
anonymous
  • anonymous
3
anonymous
  • anonymous
Good.
anonymous
  • anonymous
\[ \int^{2}_{-1}2x\;dx = (2)^2-(-1)^2=3 \]
anonymous
  • anonymous
Thank you
anonymous
  • anonymous
Medal?
anonymous
  • anonymous
Yes, thank you

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