₀^ a (a-x)^ 3 2 x² d x=

Step-by-step solution

Let aligned I &= ₀^ a (a-x)^ 3 2 x² d x &= ₀^ a [a-(a-x)]^ 3 2 (a-x)² d x &= ₀^ a x^ 3 2 (a²-2 a x+x² ) d x= ₀^ a (a² x^ 3 2 -2 a x^ 5 2 +x^ 7 2 )_ d x &=a² [ x^ 5 2 5 2 ]₀^ a -2 a [ x^ 7 2 7 2 ]₀^ a + [ x^ 9 2 9 2 ]₀^ a &= 2 a² 5 [a^ 5 2 -0 ]-2 a 2 7 [a^ 7 2 -0 ]+ 2 9 [a^ 9 2 -0 ]= 2 5 a^ 9 2 - 4 7 a^ 9 2 + 2 9 a^ 9 2 &=a^ 9 2 [ 2 5 - 4 7 + 2 9 ]= 16 315 a^ 9 2 aligned