`int _(-pi//2)^(pi//2) sin^(4) x cos^(6) x dx` is equal to

Options

A. `(3pi)/64`
B. `(3pi)/572`
C. `(3pi)/256`
D. `(3pi)/128`

Explanation

To solve the integral \( I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^4 x \cos^6 x \, dx \), we can use the property of definite integrals and symmetry.

### Step 1: Use symmetry of the integrand
The function \( \sin^4 x \cos^6 x \) is an even function. Therefore, we can simplify the integral as follows:
\[ I = 2 \int_{0}^{\frac{\pi}{2}} \sin^4 x \cos^6 x \, dx \]

### Step 2: Use the integral property
We can use the property of integrals of the form \( \int_0^{\frac{\pi}{2}} \sin^m x \cos^n x \, dx \):
\[ \int_0^{\frac{\pi}{2}} \sin^m x \cos^n x \, dx = \frac{1}{2} \cdot \frac{\Gamma\left(\frac{m+1}{2}\right) \Gamma\left(\frac{n+1}{2}\right)}{\Gamma\left(\frac{m+n+2}{2}\right)} \]
For our case, \( m = 4 \) and \( n = 6 \).

### Step 3: Calculate the Gamma functions
Now we need to calculate the Gamma functions:
- \( \Gamma\left(\frac{4+1}{2}\right) = \Gamma\left(\frac{5}{2}\right) \)
- \( \Gamma\left(\frac{6+1}{2}\right) = \Gamma\left(\frac{7}{2}\right) \)
- \( \Gamma\left(\frac{4+6+2}{2}\right) = \Gamma(6) \)

Using the property of the Gamma function:
- \( \Gamma\left(\frac{5}{2}\right) = \frac{3}{2} \Gamma\left(\frac{3}{2}\right) = \frac{3}{2} \cdot \frac{\sqrt{\pi}}{2} = \frac{3\sqrt{\pi}}{4} \)
- \( \Gamma\left(\frac{7}{2}\right) = \frac{5}{2} \Gamma\left(\frac{5}{2}\right) = \frac{5}{2} \cdot \frac{3\sqrt{\pi}}{4} = \frac{15\sqrt{\pi}}{8} \)
- \( \Gamma(6) = 5! = 120 \)

### Step 4: Substitute into the integral formula
Now we substitute these values into the integral formula:
\[ \int_0^{\frac{\pi}{2}} \sin^4 x \cos^6 x \, dx = \frac{1}{2} \cdot \frac{\frac{3\sqrt{\pi}}{4} \cdot \frac{15\sqrt{\pi}}{8}}{120} \]

### Step 5: Simplify the expression
Calculating the numerator:
\[ \frac{3\sqrt{\pi}}{4} \cdot \frac{15\sqrt{\pi}}{8} = \frac{45\pi}{32} \]
Now substituting back:
\[ \int_0^{\frac{\pi}{2}} \sin^4 x \cos^6 x \, dx = \frac{1}{2} \cdot \frac{\frac{45\pi}{32}}{120} = \frac{45\pi}{64 \cdot 120} = \frac{45\pi}{7680} \]

### Step 6: Final value of the integral
Now, recall that \( I = 2 \int_0^{\frac{\pi}{2}} \sin^4 x \cos^6 x \, dx \):
\[ I = 2 \cdot \frac{45\pi}{7680} = \frac{90\pi}{7680} = \frac{3\pi}{256} \]

Thus, the final answer is:
\[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^4 x \cos^6 x \, dx = \frac{3\pi}{256} \]