Question

Evaluate \[ \left( \int_0^1 x^4 (1 - x)^5 \, dx \right)^{-1}. \]

Hint:

The Beta function is often useful for integrals of the form \( \int_0^1 x^m (1 - x)^n \, dx \).

Answer 1

Correct Answer: 1259.9 - 1260.1

Step 1: Recognizing the beta integral.
The given integral is of the form \( \int_0^1 x^m (1 - x)^n \, dx \), which is a Beta function: \[ \int_0^1 x^m (1 - x)^n \, dx = B(m+1, n+1) = \frac{m! n!}{(m+n+1)!}. \]
Step 2: Applying the formula.
In our case, \( m = 4 \) and \( n = 5 \), so the integral becomes: \[ \int_0^1 x^4 (1 - x)^5 \, dx = B(5, 6) = \frac{4! 5!}{(4+5+1)!} = \frac{24 \times 120}{10!}. \] We compute: \[ \frac{24 \times 120}{10!} = \frac{2880}{3628800} = \frac{1}{1260}. \]
Step 3: Final result.
Thus, the reciprocal of this value is: \[ \left( \int_0^1 x^4 (1 - x)^5 \, dx \right)^{-1} = 1260. \]