Integrating --> which statement is true?

[Baron]

Please look at the attachment for the question.

My work:

I'm completely lost as to how to approach the problem. I know e^[(-t)^2] cannot be integrated so I don't know how to begin.

Any help would be appreciated.

 

[Unknown008]

Well, one thing you can be sure of, is that A and D cannot be correct, since y(0) in the above equation will result in a limit of 2 to 0, which will mean that you have 5 minus something, whereas if y(1) = 5 is true, you get the limits in the equation as 2 to 2, which is automatically 0, and y(1) = 5 + 0 = 5.

I don't know about the derivative... maybe try the other way round, that is form a differential equation with the given conditions, that is y(1) = 5.

Okay, now, I'm not sure at all, but you should have:

\(\displaystyle \displaystyle \int^y_5 y\ dy = \int^x_1 e^{-x^2} dx\)

or

\(\displaystyle \displaystyle \int^y_5 y\ dy = \int^x_1 e^{-4x^2} dx\)

or

\(\displaystyle \displaystyle \int^y_5 y\ dy = \int^x_1 2e^{-4x^2} dx\)

And I have a feeling it's the last one, why? When you substitute to put t, I think like that:

2x = t

2 dx = dt

dx = dt/2

Which puts the last one like this:

\(\displaystyle y - 5 = \int^x_1 e^{-t^2} dt\)

Then, changing the limits,

For x = 1, 2x = 2 (lower bound)
For x = x, 2x = 2x (upper bound)

 

[pka]

\(\displaystyle y = \int\limits_0^{f(x)} {g(t)dt} \, \Rightarrow \,\frac{{dy}}{{dx}} = f'(x)g(f(x))\)

 

[burakaltr]

Please look at the attachment for the question.

My work:

I'm completely lost as to how to approach the problem. I know e^[(-t)^2] cannot be integrated so I don't know how to begin.

Any help would be appreciated.

After applying Riemann Sum and the negligibility of e**(-4)= zero

you will see that

"E"

is the answer



PS : y(1)=5