Find the integral of a function having intervals 1 to 0 and the function f(x)=x^7/2.(1-x)^5/2dx

To find the integral of the function \( f(x) = x^{7/2} (1-x)^{5/2} \) over the interval from 0 to 1, we can utilize the properties of the Beta function, which is particularly useful for integrals of this form. The Beta function is defined as:

Understanding the Beta Function

The Beta function, denoted as \( B(x, y) \), is given by the integral:

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

This function relates closely to the Gamma function, where \( B(x, y) = \frac{\Gamma(x) \Gamma(y)}{\Gamma(x+y)} \). In our case, we can rewrite our integral in a form that resembles the Beta function.

Setting Up the Integral

We need to express our integral in terms of the Beta function. The integral we want to evaluate is:

\( I = \int_0^1 x^{7/2} (1-x)^{5/2} dx \)

By comparing this with the Beta function, we can identify \( x = \frac{9}{2} \) and \( y = \frac{7}{2} \). Thus, we can express our integral as:

\( I = B\left(\frac{9}{2}, \frac{7}{2}\right) \)

Calculating the Beta Function

Now, we can use the relationship between the Beta function and the Gamma function:

\( B\left(\frac{9}{2}, \frac{7}{2}\right) = \frac{\Gamma\left(\frac{9}{2}\right) \Gamma\left(\frac{7}{2}\right)}{\Gamma\left(\frac{16}{2}\right)} = \frac{\Gamma\left(\frac{9}{2}\right) \Gamma\left(\frac{7}{2}\right)}{\Gamma(8)} \)

Finding the Gamma Values

Next, we need to compute the Gamma functions:

• Using the property \( \Gamma(n + 1) = n \Gamma(n) \), we find:
• For \( \Gamma\left(\frac{9}{2}\right) = \frac{7}{2} \Gamma\left(\frac{7}{2}\right) \)
• For \( \Gamma\left(\frac{7}{2}\right) = \frac{5}{2} \Gamma\left(\frac{5}{2}\right) \)
• And \( \Gamma(8) = 7! = 5040 \)

Calculating the Values

We know that:

• \( \Gamma\left(\frac{5}{2}\right) = \frac{3\sqrt{\pi}}{4} \)
• Thus, \( \Gamma\left(\frac{7}{2}\right) = \frac{5}{2} \cdot \frac{3\sqrt{\pi}}{4} = \frac{15\sqrt{\pi}}{8} \)
• And \( \Gamma\left(\frac{9}{2}\right) = \frac{7}{2} \cdot \frac{15\sqrt{\pi}}{8} = \frac{105\sqrt{\pi}}{16} \)

Putting It All Together

Now we can substitute these values back into our expression for the Beta function:

\( B\left(\frac{9}{2}, \frac{7}{2}\right) = \frac{\frac{105\sqrt{\pi}}{16} \cdot \frac{15\sqrt{\pi}}{8}}{5040} = \frac{1575\pi}{128 \cdot 5040} \)

Calculating the denominator, \( 128 \cdot 5040 = 645120 \), we find:

\( I = \frac{1575\pi}{645120} \)

Final Result

Thus, the integral of the function \( f(x) = x^{7/2} (1-x)^{5/2} \) from 0 to 1 is:

\( \int_0^1 x^{7/2} (1-x)^{5/2} dx = \frac{1575\pi}{645120} \)

This result can be simplified further if needed, but it provides a clear answer to the integral we set out to solve.