How do you Use the trapezoidal rule with n=10n=10 to approximate the integral int_1^2ln(x)/(1+x)dx21ln(x)1+xdx?

1 Answer
Sep 6, 2014

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.