use simpsons rule with n = 4 to approximate ∫₁⁵ (cos(x)/x) dx keep at least 2 decimal places accuracy in…

Explanation:

Step1: Calculate the step - size $h$

$a = 1$, $b = 5$, $n = 4$. Then $h=\frac{b - a}{n}=\frac{5 - 1}{4}=1$.

Step2: Define the function $f(x)$

$f(x)=\frac{\cos(x)}{x}$.

Step3: Calculate the values of $f(x)$ at the endpoints and mid - points

$x_0 = 1$, $f(x_0)=\frac{\cos(1)}{1}\approx0.5403$; $x_1 = 2$, $f(x_1)=\frac{\cos(2)}{2}\approx - 0.2081$; $x_2 = 3$, $f(x_2)=\frac{\cos(3)}{3}\approx - 0.1667$; $x_3 = 4$, $f(x_3)=\frac{\cos(4)}{4}\approx - 0.1655$; $x_4 = 5$, $f(x_4)=\frac{\cos(5)}{5}\approx0.0567$.

Step4: Apply Simpson's rule

$S_n=\frac{h}{3}[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+f(x_4)]$ $S_4=\frac{1}{3}[0.5403 + 4\times(- 0.2081)+2\times(- 0.1667)+4\times(- 0.1655)+0.0567]$ $S_4=\frac{1}{3}[0.5403-0.8324 - 0.3334-0.662+0.0567]$ $S_4=\frac{1}{3}[0.5403 + 0.0567-(0.8324 + 0.3334+0.662)]$ $S_4=\frac{1}{3}[0.597 - 1.8278]$ $S_4=\frac{1}{3}\times(-1.2308)\approx - 0.41$.

Answer:

$-0.41$