grandes-ecoles 2013 QIII.D.3

grandes-ecoles · France · centrale-maths1__pc Reduction Formulae Establish an Integral Identity or Representation
Deduce that $$\int _ { 0 } ^ { 2 \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) d \theta = \begin{cases} 4 \pi \ln ( | x | ) & \text { if } | x | > 1 \\ 0 & \text { if } | x | < 1 \end{cases}$$ 
Deduce that
$$\int _ { 0 } ^ { 2 \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) d \theta = \begin{cases} 4 \pi \ln ( | x | ) & \text { if } | x | > 1 \\ 0 & \text { if } | x | < 1 \end{cases}$$
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