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

Question

Ezekiel Alexander · Accepted Answer

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)))# 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. 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)))# 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# 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.