Question

Evaluate the integral \(\int\limits_0^{\frac{\pi }{2}} {{{\sin }^7}\theta {{\cos }^5}\theta d\theta } \).

Short answer

\(\frac{1}{{120}}\) is the Answer to given solution

Step-by-step solution

Step-1: Wallis Formula

\(\begin{aligned}{l}\int\limits_0^{\frac{n}{2}} {{{\sin }^n}n{{\cos }^m}ndn} \left( {m,n \in n} \right)\\\end{aligned}\)

\(\left( {\frac{{(n - 10)(n - 3)...(1or2)...(m - 1)(m - 3)...1or2}}{{(m + n)(m + n + 2)....1or2}}} \right)\)

Step-2: Calculation of Integral

\(\begin{aligned}{l}\int\limits_0^{\frac{\pi }{2}} {{{\sin }^7}\theta {{\cos }^5}\theta d\theta } \\ &= \frac{{(7 - 1)(7 - 3)(7 - 5)...(5 - 1)(5 - 3)}}{{12 \times (12 - 2)(12 - 4)(12 - 6)(12 - 8)(12 - 10)}}\\ &= \frac{{6 \times 4 \times 2 \times 4 \times 2}}{{12 \times 10 \times 8 \times 6 \times 4 \times 2}}\\ &= \frac{1}{{120}}\end{aligned}\)

Hence, \(\frac{1}{{120}}\) this is the required answer.