# Thread: Calculating the following integrals

1. ## Calculating the following integrals

Hullo,

I need some help calculating the following integrals:

$\lim_{x\to 0}\; \dfrac{1}{x} \int_0^x\; e^{t^2}\; dt
$

and this one:

$\lim_{h\to 0}\; \dfrac{1}{h} \int_3^{3+h}\; e^{t^2}\; dt
$

using the FTC.

Any help would be appreciated.

-the Doctor

2. Originally Posted by thedoctor818
Hullo,

I need some help calculating the following integrals:

$\lim_{x\to 0}\; \dfrac{1}{x} \int_0^x\; e^{t^2}\; dt
$

and this one:

$\lim_{h\to 0}\; \dfrac{1}{h} \int_3^{3+h}\; e^{t^2}\; dt
$

using the FTC.

Any help would be appreciated.

-the Doctor
Both these problems have very similar solutions.

Here's a way to start

Let $F(x) = \int_0^{x}\; e^{t^2}\; dt$

Notice that (for part 2) $\frac{F(x+h) - F(x)}{h} = \frac{\int_0^{x+h}\; e^{t^2}\; dt - \int_0^{x}\; e^{t^2}\; dt}{h} = \frac{1}{h}\int_x^{x+h}\; e^{t^2}\; dt$...

3. I am sorry, but am still a bit confused. So, for the first integral, am I just taking $lim_{x\to0}\dfrac{F(x)}{x} ?$ If so , how do I evaluate since there is no 'elementary' antiderivative. And in the second, am I taking $lim_{h\to0}\dfrac{F(x+h)-F(x)}{h} ?$

Sorry, but I am still somewhat confused.

-Michael

4. Originally Posted by thedoctor818
And in the second, am I taking $lim_{h\to0}\dfrac{F(x+h)-F(x)}{h} ?$
Yes, that should look like something familiar.

So, for the first integral, am I just taking
Yes, but note that F(0) = 0, so we can rewrite it as:

$\dfrac{F(x)}{x} = \dfrac{F(x)-F(0)}{x-0}$

5. Thanks a bunch - that really helped out.

-Michael