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.

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. $e^{x+1}$

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. $e^{x^2}$.

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.