Prove that \(\int _ { 0 } ^ { N } \ln ( 1 + x ) \mathrm { d } x = ( N + 1 ) \ln ( N + 1 ) - N\), where \(N\) is a positive constant.
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The diagram shows the curve \(y = \ln ( 1 + x )\), for \(0 \leqslant x \leqslant 70\), together with a set of rectangles of unit width.
(a) By considering the areas of these rectangles, explain why
$$\ln 2 + \ln 3 + \ln 4 + \ldots + \ln 70 < \int _ { 0 } ^ { 70 } \ln ( 1 + x ) d x$$
(b) By considering the areas of another set of rectangles, show that
$$\ln 2 + \ln 3 + \ln 4 + \ldots + \ln 70 > \int _ { 0 } ^ { 69 } \ln ( 1 + x ) d x$$
(c) Hence find bounds between which \(\ln ( 70 ! )\) lies. Give the answers correct to 1 decimal place.