Evaluate the following :
`int(x-(1)/(x))^(5//2)((x^2+1)/(x^2)) dx`

Explanation

(i) `I=int(x+(1)/(x))^(3//2)((x^(2)-1)/(x^(2)))dx=int(x+(1)/(x))^(3//2)(1-(1)/(x^(2)))dx`
` "let " t=x+(1)/(x) impliesdt=(1-(1)/(x^(2)))dx`
`impliesI=int t^(3//2)dt=(2)/(5)t^(5//2)+C=(2)/(5)(x+(1)/(x))^(5//2)+C`
(ii) ` int(sqrt(2+logx))/(x)dx=int(2+logx)^((1)/(2))(d)/(dx)(2+logx)dx`
`=((2+logx)^((3)/(2)))/((3)/(2))+c`
`=(2(2+logx)^((3)/(2)))/(3)+c`
(iii) `int((sin^(-1)x)^(3))/(sqrt(1-x^(2)))dx=int(sin^(-1)x)^(3)(d)/(dx)(sin^(-1)x)dx`
`=((sin^(-1)x)^(4))/(4)+c`
(iv) `int(cotx)/(sqrt(sinx))dx=int(cosx)/(sin x sqrt(sinx))dx`
`=int(sinx)^(-3//2)cos xdx`
`=((sinx)^(-1//2))/(-1//2)+c`
`=(-2)/(sqrt(sinx))+c`