Question

Assuming the step size \( h = 1 \), the numeric value (rounded off to 2 decimal places) of the definite integral
\[ \int_1^3 \frac{x}{1+x} \, dx \] obtained using Simpson’s rule is ________________.

Hint:

Simpson's rule works best when the function is smooth and continuous over the interval.

Answer 1

Correct Answer: 1.29

We need to approximate the integral using Simpson’s Rule:
\[ \int_1^3 \frac{x}{1+x}\,dx \]
Simpson’s rule formula is:
\[ \int_a^b f(x)dx \approx \frac{h}{3}[f(x_0)+4f(x_1)+f(x_2)] \]
Here, \(a =1\), \(b=3\), \(h=1\), and the points are \(x_0 =1\), \(x_1 =2\), \(x_2 =3\).
Now compute function values:
\(f(1)= \frac{1}{2} = 0.5\)
\(f(2)= \frac{2}{3}\)
\(f(3)= \frac{3}{4} = 0.75\)
Apply Simpson’s rule:
\[ \frac{1}{3}\left(0.5 + 4\cdot\frac{2}{3} + 0.75\right) \]
\[ = \frac{1}{3}(0.5 + 2.6667 + 0.75) \]
\[ = \frac{3.9167}{3} = 1.3056 \]
Rounded to two decimals, this is 1.31.
Final Answer: 1.31