Use the trapezium rule, with two strips of equal width, to show that
$$\int _ { 0 } ^ { 4 } \frac { 1 } { 2 + \sqrt { x } } \mathrm {~d} x \approx \frac { 11 } { 4 } - \sqrt { 2 }$$
Use the substitution \(x = u ^ { 2 }\) to find the exact value of
$$\int _ { 0 } ^ { 4 } \frac { 1 } { 2 + \sqrt { x } } \mathrm {~d} x$$
Using your answers to parts (i) and (ii), show that
$$\ln 2 \approx k + \frac { \sqrt { 2 } } { 4 }$$
where \(k\) is a rational number to be determined.