grandes-ecoles 2015 QIV.B.2

grandes-ecoles · France · centrale-maths2__mp Reduction Formulae Prove Regularity or Structural Properties of an Integral-Defined Function
We denote $B$ the function defined on $\mathbb { R } ^ { + * }$ by $B ( x ) = \int _ { 0 } ^ { 1 } ( \ln ( 1 - t ) ) ^ { 2 } t ^ { x - 1 } \mathrm {~d} t$.
Give without justification an expression, using an integral, of $B ^ { ( p ) } ( x )$, for every natural integer $p$ and every real $x > 0$. 
We denote $B$ the function defined on $\mathbb { R } ^ { + * }$ by $B ( x ) = \int _ { 0 } ^ { 1 } ( \ln ( 1 - t ) ) ^ { 2 } t ^ { x - 1 } \mathrm {~d} t$.

Give without justification an expression, using an integral, of $B ^ { ( p ) } ( x )$, for every natural integer $p$ and every real $x > 0$.
Paper Questions
QI.A.1 QI.A.2 QI.B QI.C.1 QI.C.2 QI.D.1 QI.D.2 QI.D.3 QI.D.4 QI.E QI.F.1 QI.F.2 QI.F.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 QII.C.4 QII.C.5 QII.C.6 QII.C.7 QII.C.8 QIII.A QIII.B.1 QIII.B.2 QIII.B.3 QIII.C.1 QIII.C.2 QIII.C.3 QIII.D.1 QIII.D.2 QIII.D.3 QIV.A QIV.B.1 QIV.B.2 QIV.B.3 QIV.B.4 QIV.C.1 QIV.C.2