Question

How do you Use the trapezoidal rule with `n=10` to approximate the integral `int_1^2ln(x)/(1+x)dx`?

Answer 1

The trapezoidal rule is:
`int_a^bf(x)dx ~~ (b-a)/(2n)*(f(x_1)+2f(x_2)+...+2f(x_n)+f(x_(n+1)))`

1. First, we need to find our different values of x: since `1<=x<=2` and we need to split up our function into ten parts. So, quite simply, the values of x we need are 1, 1.1, 1.2, ..., 1.9, 2, with `x_1=1`, `x_2=1.1`, and so on.

2. Next, we need to substitute the values of `a` and `b`, which are 1 and 2 respectively, into the equation. Also plug in `n=10`.
   `int_1^2ln(x)/(1+x)dx ~~ (2-1)/(2*10)*(f(x_1)+2f(x_2)+...+2f(x_10)+f(x_(11)))`

3. Next, we evaluate `f(x_1)`, `f(x_2)` and so on. As an example:
   `f(x_1) = f(1) = ln(1)/(1+1) = ln(1)/2 = 0`
   `f(x_2) = f(1.1) = ln(1.1)/(1+1.1) = ln(1.1)/2.1 ~~ 0.04539`

4. Finally, we plug all these values into our equation.
   `int_1^2ln(x)/(1+x)dx ~~ (2-1)/(2*10)*(f(1)+2f(1.1)+...)`

This should provide an approximate answer to any integral.