[size=150]The Wallis integral formula is an integral representation of the Wallis product used to calculate the value of [math]\pi[/math]. The Wallis integral formula is given by:[br][br][math]\frac{\pi}{2}=\int cos^n\left(x\right)dx[/math]; from 0 to [math]\frac{\pi}{2}[/math][br][br]where n is a positive integer.[br][br]To calculate [math]\pi[/math], we can use the Wallis integral formula with a specific value of n and then solve the integral. Let's consider n = 4 for this example:[br][br][math]\frac{\pi}{2}=cos^4\left(x\right)dx[/math]; for 0 to [math]\frac{\pi}{2}[/math][br][br][math][/math][br]To solve the integral, we can use trigonometric identities. Since [math]cos^2\left(x\right)=\frac{1}{2}\left(1+cos\left(2x\right)\right)[/math], we can rewrite [math]cos^4\left(x\right)[/math] as:[br][br][math]cos^4\left(x\right)=\left(\frac{1}{2}\left(1+cos\left(2x\right)\right)\right)^2=\frac{1}{4}\left(1+2cos\left(2x\right)\right)+cos^2\left(2x\right)[/math][br][br]Now, we integrate [math]cos^4\left(x\right)[/math] with respect to x over the interval [math]\left[0,\frac{\pi}{2}\right][/math][br][br][math]\frac{\pi}{2}=\int\frac{1}{4}\left(1+2cos\left(2x\right)\right)+cos^2\left(2x\right)dx[/math][br][br][math]\frac{\pi}{2}=\frac{1}{4}\int\left(1+2cos\left(2x\right)\right)+cos^2\left(2x\right)dx[/math][br][br][math]\frac{\pi}{2}=\frac{1}{4}\left[x+sin\left(2x\right)+\frac{1}{2}sin\left(4x\right)\right][/math]from to to [math]\frac{\pi}{2}[/math][br][br]Evaluate the integral at the upper and lower limits:[br][br][math]\frac{\pi}{2}=\frac{1}{4}\left[\frac{\pi}{2}+sin\left(x\right)+\frac{1}{2}sin\left(2\pi\right)-\left(0+sin\left(0\right)+\frac{1}{2}sin\left(0\right)\right)\right][/math][br][br][math]\frac{\pi}{2}=\frac{1}{4}\left[\frac{\pi}{2}+0+0\right][/math][br][br][math]\frac{\pi}{2}=\frac{\pi}{8}[/math][br][br]Finally, to find the value of [math]\pi[/math], we multiply both sides by 2:[br][br][math]\frac{\pi}{2}=2\cdot\frac{\pi}{8}=\frac{\pi}{4}[/math][br][br]So, using the Wallis integral formula with n = 4, we find that [math]\pi[/math][math]=\frac{\pi}{4}[/math]. This is an incorrect result, which means that the Wallis integral formula with n = 4 does not give the correct value of [math]\pi[/math]. However, as n approaches infinity in the Wallis integral formula, it converges to the correct value of [math]\frac{\pi}{2}[/math].[/size]