here is an example from and old paper i've found-

(a) use the trapezium rule with five ordinates to find an approximate value for

integrate[log(1+X^2)(0 min)(1 max)]

give answer to 2 d.p

answer im pretty sure is 0.09

(b) use your answer to part a to deduce an appropriate an approximate value for

integrate[log(sqrt(1+X^2))(0 min)(1 max)]

any workings needed i have

Ah, I didn't realize the integrand changed so much.

I was stumped by this for a bit but then realized this is a simple application of logarithm rules.

\(\displaystyle \int_0^1\log(\sqrt{x^2+1})\,dx\)

\(\displaystyle =\int_0^1\log\left((x^2+1)^{\frac{1}{2}}\right)\,dx\)

\(\displaystyle =\int_0^1\frac{1}{2}\cdot\log(x^2+1)\,dx\)

\(\displaystyle =\frac{1}{2}\cdot\int_0^1\log(x^2+1)\,dx\)

By the way, assuming log means natural log, I get 0.27 as my approximation for part (a).