6 The equation of a curve is \(y = \frac { 1 } { 1 + \tan x }\).
- Show, by differentiation, that the gradient of the curve is always negative.
- Use the trapezium rule with 2 intervals to estimate the value of
$$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { 1 } { 1 + \tan x } \mathrm {~d} x$$
giving your answer correct to 2 significant figures.
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The diagram shows a sketch of the curve for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\). State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the integral in part (ii).