Evaluate : `intx^(3)sqrt(x^(2)-4) dx`

Explanation

`"Let I " =int x^(3)sqrt(x^(2)-4)dx`
`=int x^(2)sqrt(x^(2)-4).(x.dx)`
`" Let "x^(2) -4=t`
`rArr " " 2x =(dt)/(dx)`
`rArr " "x.dx =(dt)/(2)`
`:. I=int(t+4)sqrt(t.)(dt)/(2) =1/2int(t^(3//2)+4t^(1//2))dt`
`=1/2[(2)/(5)t^(5//2)+(8)/(3)t^(3//2)]+c`
`=1/5(x^(2)-4)^(5//2)+4/5(x^(2)-4)^(3//2)+c`