Evaluate the following :
`int_(0)^(a)x^(2)(a-x)^((3)/(2))dx.`

Explanation

We use the property, `int_(0)^(a)f(x)dx=int_(0)^(a)f(a-x)dx`
`therefore int_(a)^(a)x^(2)(a-x)^((3)/(2))dx=int_(0)^(a)(a-x)^(2)(a-x+x)^((3)/(2))dx`
`=int_(0)^(a)(a^(2)-2ax+x^(2))x^((3)/(2))dx=int_(0)^(a)(a^(2)x^((3)/(2))-2ax^((5)/(2))+x^((7)/(2)))dx`
`=[a^(2).(x^((5)/(2)))/(5//2)-2a.(x^((7)/(2)))/(7//2)+(x^((9)/(2)))/(9//2)]_(0)^(a)`
`=(2a^(2))/(5).(a)^((5)/(2))-(4a)/(7).(a)^((7)/(2))+(2)/(9).(a)^((9)/(2))-0`
`=(2)/(5)a^((9)/(2))-(4)/(7)a^((9)/(2))+(2)/(9)a^((9)/(2))=((2)/(5)-(4)/(7)+(2)/(9))a^((9)(2))`
`=((126-180+70)/(315))a^((9)/(a))=(16a^((9)/(2)))/(315).`