To evaluate the integral of \( \exp(-x^2) \) between 0 and 1, we can use the error function (\(\text{erf}(x)\)), which is related to the Gaussian integral.

The integral in question is:
\[ \int_0^1 e^{-x^2} \, dx \]

The indefinite integral of \( e^{-x^2} \) does not have an elementary antiderivative, but it can be expressed in terms of the error function, \(\text{erf}(x)\), which is defined as:
\[ \text{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \, dt \]

Using the error function, the definite integral can be written as:
\[ \int_0^1 e^{-x^2} \, dx = \frac{\sqrt{\pi}}{2} \left( \text{erf}(1) - \text{erf}(0) \right) \]

We know that \(\text{erf}(0) = 0\). Therefore:
\[ \int_0^1 e^{-x^2} \, dx = \frac{\sqrt{\pi}}{2} \text{erf}(1) \]

The value of \(\text{erf}(1)\) is a known constant:
\[ \text{erf}(1) \approx 0.842700792949715 \]

Thus, the integral becomes:
\[ \int_0^1 e^{-x^2} \, dx \approx \frac{\sqrt{\pi}}{2} \times 0.842700792949715 \]

Approximating \(\sqrt{\pi} \approx 1.77245385091\):
\[ \int_0^1 e^{-x^2} \, dx \approx \frac{1.77245385091}{2} \times 0.842700792949715 \]
\[ \int_0^1 e^{-x^2} \, dx \approx 0.747858 \]

So, the integral of \( \exp(-x^2) \) between 0 and 1 is approximately \( 0.747858 \).

Would you like more details or have any questions?

Here are five related questions to expand your understanding:

1. How would you evaluate the integral \( \int_0^1 e^{-x} \, dx \)?
2. What is the value of \( \int_{-\infty}^\infty e^{-x^2} \, dx \)?
3. How do you evaluate \( \int_0^2 e^{-x^2} \, dx \)?
4. What is the relationship between the error function and the integral of \( e^{-x^2} \)?
5. How would you approximate the integral \( \int_0^1 e^{-x^2} \, dx \) using numerical methods?

**Tip:** When dealing with integrals of functions that do not have elementary antiderivatives, consider using special functions like the error function or numerical integration techniques.

Learn how to evaluate the integral of exp(-x^2) from 0 to 1 using the error function. This problem covers integral calculus, special functions like the error function (erf), and their applications. Suitable for advanced high school students and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation