https://file.aiquickdraw.com/mathbot/file/1727072934906d4jgily3.jpg

The integral shown in the image is:

\[
\int_0^1 t^4 (1 - t)^3 \, dt
\]

This type of integral can be evaluated using the **Beta function**. The Beta function \( B(x, y) \) is defined as:

\[
B(x, y) = \int_0^1 t^{x-1} (1 - t)^{y-1} \, dt
\]

By comparing the given integral with the form of the Beta function, we can identify:

\[
t^4 = t^{5-1}, \quad (1 - t)^3 = (1 - t)^{4-1}
\]

So the integral matches the form of the Beta function with \( x = 5 \) and \( y = 4 \). Therefore, we can express the integral as:

\[
\int_0^1 t^4 (1 - t)^3 \, dt = B(5, 4)
\]

The Beta function is related to the Gamma function \( \Gamma(n) \) by the identity:

\[
B(x, y) = \frac{\Gamma(x) \Gamma(y)}{\Gamma(x + y)}
\]

Using this identity, we can compute \( B(5, 4) \):

\[
B(5, 4) = \frac{\Gamma(5) \Gamma(4)}{\Gamma(9)}
\]

Now, using the fact that \( \Gamma(n) = (n-1)! \) for positive integers:

\[
\Gamma(5) = 4!, \quad \Gamma(4) = 3!, \quad \Gamma(9) = 8!
\]

So:

\[
B(5, 4) = \frac{4! \times 3!}{8!} = \frac{24 \times 6}{40320} = \frac{144}{40320} = \frac{1}{280}
\]

Thus, the value of the integral is:

\[
\int_0^1 t^4 (1 - t)^3 \, dt = \frac{1}{280}
\]

Would you like further details or have any questions?

### Relative Questions:
1. What is the relationship between the Beta and Gamma functions?
2. How do you calculate the Gamma function for non-integer values?
3. What are some applications of the Beta function in probability and statistics?
4. Can this integral be solved using any other method apart from the Beta function?
5. How does the Beta function generalize to higher dimensions?

### Tip:
The Beta function can be especially useful in evaluating integrals that arise in probability distributions, such as the Beta distribution.

Learn how to solve the integral ∫_0^1 t^4(1 - t)^3 dt using the Beta function, involving concepts from advanced calculus. The problem highlights the relationship between the Beta and Gamma functions. Suitable for undergraduate students studying calculus or mathematical analysis.

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