grandes-ecoles 2013 QIII.D.1

grandes-ecoles · France · centrale-maths1__pc Indefinite & Definite Integrals Definite Integral as a Limit of Riemann Sums
Show that $\forall x \in \mathbb { R } \backslash \{ - 1,1 \}$ $$\int _ { 0 } ^ { 2 \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta = \lim _ { n \rightarrow + \infty } \left( \frac { 2 \pi } { n } \sum _ { k = 1 } ^ { n } \ln \left( x ^ { 2 } - 2 x \cos \frac { 2 k \pi } { n } + 1 \right) \right)$$ 
Show that $\forall x \in \mathbb { R } \backslash \{ - 1,1 \}$
$$\int _ { 0 } ^ { 2 \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta = \lim _ { n \rightarrow + \infty } \left( \frac { 2 \pi } { n } \sum _ { k = 1 } ^ { n } \ln \left( x ^ { 2 } - 2 x \cos \frac { 2 k \pi } { n } + 1 \right) \right)$$
Paper Questions
QI.A.1 QI.A.2 QI.A.3 QI.B.1 QI.B.2 QI.C.1 QI.C.2 QI.C.3 QII.A.1 QII.A.2 QII.B.1 QII.B.2 QII.B.3 QII.B.4 QII.C.1 QII.C.2 QII.C.3 QIII.A.1 QIII.A.2 QIII.A.3 QIII.A.4 QIII.B.1 QIII.B.2 QIII.B.3 QIII.B.4 QIII.B.5 QIII.B.6 QIII.C.1 QIII.C.2 QIII.C.3 QIII.C.4 QIII.D.1 QIII.D.2 QIII.D.3 QIII.D.4 QIII.D.5 QIII.D.6 QIII.D.7 QIV.A.1 QIV.A.2 QIV.A.3 QIV.A.4 QIV.B QIV.C.1 QIV.C.2