Integral Calculation for \(\mathop \smallint \nolimits_0^{\frac{\pi }{2}} {\sin ^7}xdx\)

The problem requires finding the value of the definite integral of the function \(f(x) = {\sin ^7}x\) from 0 to \(\frac{\pi }{2}\). This type of integral can be efficiently solved using a standard formula known as Walli's Integral.

Walli's Integral Formula Application

Walli's Integral provides a formula for definite integrals of the form:

• $$\mathop \smallint \nolimits_0^{\frac{\pi }{2}} {\sin ^n}x\,dx = \mathop \smallint \nolimits_0^{\frac{\pi }{2}} {\cos ^n}x\,dx$$

The formula depends on whether the power \(n\) is even or odd:

• If \(n\) is even: The result is $$\frac{(n-1)!!}{n!!} \times \frac{\pi}{2}$$
• If \(n\) is odd: The result is $$\frac{(n-1)!!}{n!!}$$

Here, \(n\) represents the power of the sine (or cosine) function. The double factorial (!!) means multiplying integers decreasing by 2 until 1 or 2 is reached. For example, \(5!! = 5 \times 3 \times 1\) and \(6!! = 6 \times 4 \times 2\).

Step-by-Step Calculation

Let's apply Walli's Integral formula to the given integral:

1. Identify the power \(n\): In the integral \(\mathop \smallint \nolimits_0^{\frac{\pi }{2}} {\sin ^7}xdx\), the power \(n\) is 7.
2. Determine if \(n\) is even or odd: Since \(n=7\), it is an odd number.
3. Apply the formula for odd \(n\): The formula is $$\frac{(n-1)!!}{n!!}$$.
4. Substitute \(n=7\) into the formula: This gives $$\frac{(7-1)!!}{7!!} = \frac{6!!}{7!!}$$.
5. Calculate the double factorials:
   • \(6!! = 6 \times 4 \times 2 = 48\)
   • \(7!! = 7 \times 5 \times 3 \times 1 = 105\)

   So the value is $$\frac{48}{105}$$.

6. Simplify the fraction: Both the numerator (48) and the denominator (105) are divisible by 3.
   • $$ \frac{48 \div 3}{105 \div 3} = \frac{16}{35} $$

Therefore, the value of the definite integral \(\mathop \smallint \nolimits_0^{\frac{\pi }{2}} {\sin ^7}xdx\) is \(\frac{16}{35}\).