Approximate the integral using Simpson's Rule: $\int _ { 0 } ^ { 5 } \frac { x } { 2 + x + x ^ { 2 } } d x$ , n = 6. Round your answer to three decimal places.
A)0.652
B)0.850
C)1.161
D)1.017
E)1.284

Question

Quizplus · Accepted Answer

Simpson's Rule is given by the formula $\frac{\Delta x}{3}[f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \cdots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n)]$, where $\Delta x = \frac{b-a}{n}$ and $n$ is even. Here, $a = 0$, $b = 5$, and $n = 6$, so $\Delta x = \frac{5-0}{6} = \frac{5}{6}$. The function is $f(x) = \frac{x}{2+x+x^2}$. Calculating the values of $f(x)$ at the required points and applying them to Simpson's Rule formula, the approximate value of the integral is found to be 1.017.

Approximate the Integral Using Simpson's Rule ∫05x2+x+x2dx\int _ { 0 } ^ { 5 } \frac { x } { 2 + x + x ^ { 2 } } d x∫05​2+x+x2x​dx

Approximate the Integral Using Simpson's Rule $\int _ { 0 } ^ { 5 } \frac { x } { 2 + x + x ^ { 2 } } d x$