9.0 Walli's Formula

$$\int\limits_0^{{\pi \over 2}} {{{\sin }^m}x \cdot {{\cos }^n}xdx} = {{\left[ {\left( {m - 1} \right)\left( {m - 3} \right)\left( {m - 5} \right)...1or2} \right]\left[ {\left( {n - 1} \right)\left( {n - 3} \right)\left( {n - 5} \right)...1or2} \right]} \over {\left( {m + n} \right)\left( {m + n - 2} \right)\left( {m + n - 4} \right)...1or2}}K$$

When $m$ and $n$ both are even then $K = {\pi \over 2}$

else $K = 1$

If $m$ or $n$ is zero then eliminate that part of equation that include $m$ or $n$ accordingly.

As this formula is not applicable for $m,n = 0$, so formula is reduced in that condition.

Question 22. $I = \int\limits_0^{{\pi \over 2}} {{{\sin }^7}xdx} $

Solution: $m = 7$ and $n = 0$

Using Walli's formula, $$I = {{\left[ {\left( {7 - 1} \right)\left( {7 - 3} \right)2} \right]} \over {\left( 7 \right)\left( {7 - 2} \right)\left( {7 - 4} \right)}} \cdot 1$$ $$I = {{6 \cdot 4 \cdot 2} \over {7 \cdot 5 \cdot 3}}$$ $$I = {{16} \over {35}}$$

Question 23. $I = \int\limits_0^{{\pi \over 2}} {{{\sin }^8}x \cdot {{\cos }^4}xdx} $

Solution: $m = 8$ and $n = 4$ so $K = {\pi \over 2}$ $$I = {{\left[ {\left( {8 - 1} \right)\left( {8 - 3} \right)\left( {8 - 5} \right)1} \right]\left[ {\left( {4 - 1} \right)1} \right]} \over {\left( {8 + 4} \right)\left( {8 + 4 - 2} \right)\left( {8 + 4 - 4} \right)...2}}{\pi \over 2}$$ $$I = {{\left[ {7 \cdot 5 \cdot 3 \cdot 1} \right]\left[ {3 \cdot 1} \right]} \over {12 \cdot 10 \cdot 8 \cdot 6 \cdot 4 \cdot 2}}{\pi \over 2}$$ $$I = {{7\pi } \over {4 \cdot 2 \cdot 8 \cdot 2 \cdot 4 \cdot 2 \cdot 2}}$$ $$I = {{7\pi } \over {2048}}$$