grandes-ecoles 2015 QII.B.2

grandes-ecoles · France · centrale-maths2__mp Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution

For $( x , y )$ in $\left( \mathbb { R } ^ { + * } \right) ^ { 2 }$, we define $\beta ( x , y ) = \int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } \mathrm {~d} t$.
Show that for all real $x > 0$ and $y > 0 , \beta ( x , y ) = \beta ( y , x )$.

For $( x , y )$ in $\left( \mathbb { R } ^ { + * } \right) ^ { 2 }$, we define $\beta ( x , y ) = \int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } \mathrm {~d} t$.

Show that for all real $x > 0$ and $y > 0 , \beta ( x , y ) = \beta ( y , x )$.

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