https://file.aiquickdraw.com/mathbot/file/17254912126368twemve0.jpg

from PIL import Image

# Open and display the image to read the text
img_path = "/mnt/data/17254912126368twemve0.jpg"
img = Image.open(img_path)
img.show()From the uploaded image, I can see that the problem asks for the value of a given integral function. To proceed, let me extract the text and provide the solution.

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The problem in the image is:

Evaluate the integral \( f(x) = \int_0^x e^{t^2} dt \). What is \( f'(x) \)?

The options are:
1. \( 2x e^{x^2} \)
2. \( e^{x^2} \)
3. \( e^{x} \)
4. \( e^{x+1} \)

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### Solution:

Using the **Fundamental Theorem of Calculus**, we know that if:

\[
F(x) = \int_a^x f(t) \, dt
\]

Then:

\[
F'(x) = f(x)
\]

In this case, the integral given is:

\[
f(x) = \int_0^x e^{t^2} dt
\]

By the Fundamental Theorem of Calculus, the derivative of this function is simply:

\[
f'(x) = e^{x^2}
\]

Thus, the correct answer is option **2. \( e^{x^2} \)**.

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Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:
1. What is the Fundamental Theorem of Calculus and how is it applied?
2. How do you differentiate a function defined as an integral?
3. What are the applications of the function \( e^{t^2} \) in real-world problems?
4. How does the chain rule apply in cases where the upper limit of an integral is a function of \( x \)?
5. Can integrals like \( \int_0^x e^{t^2} dt \) be evaluated without numerical methods?

### Tip:
When differentiating integrals where the upper limit is a function of \( x \), always apply the Fundamental Theorem of Calculus directly.

Evaluate the integral f(x) = ∫_0^x e^{t^2} dt. What is f'(x)?

Learn how to differentiate the integral function f(x) = ∫_0^x e^{t^2} dt using the Fundamental Theorem of Calculus. This calculus problem is suitable for students in Grades 11-12 or college-level courses.

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