Evaluate:
(i) `int(5x^(2) + 2x^(-5) - 7x + 1/(sqrt(x)) +5/x) dx`
(ii) `int(3sinx - 4 cosx + 5 sec^(2) x - 2cosec^(2)x) dx`
(iii) `int(1-x)(2+3x)(5-4x)dx`
(iv) `int((3x^(4) - 5x^(3) + 4x^(2) -x + 2)/(x^(3)))dx` , (v) `int(x^(2) + 1/(x^(2)))^(3) dx`

Explanation

`int(5x^(3) + 2x^(-5) - 7x +1/(sqrt(x)) + 5/x) dx`
`= 5intx^(3)dx + 2 intx^(-5) dx - 7intx dx + intx^(-1//2) dx + 5 int(1)/(x) dx`
`= 5.(x^(4))/(4) + 2.(x^(-4))/((-4))-7.(x^(2))/(2) + (x^(1//2))/((1//2)) + 5 log |x| + C`.
`= (5x^(4))/(4) - 1/(2x^(4)) - (7x^(2))/(2) +2sqrt(x) + 5 log|x| + C`
(ii) `int(3 sinx - 4 cosx + 5sec^(2) x - 2 cosec^(2)x)dx`
`= 3int sin x dx - 4 int cos xdx + 5 int sec^(2)x dx - 2 int cosec^(2) xdx`
`= 3(-cosx) - 4sinx + 5 tanx - 2 (-cotx) + C`
`= (-3 cos x - 4sinx + 5 tan x + 2cot x + C)`.
(iii) `int(1-x) (2+3x) (5-4x) dx = int(10-3x -19x^(2) +12x^(3))dx`
`= 10intdx - 3 intx dx - 19 int x^(2)dx + 12int x^(3)dx`.
`= 10x - 3 .(x^(2))/(2) - 19.(x^(3))/(3) + 12.(x^(4))/(4) + C`.
`= 10x - (3x^(2))/(2)- (19x^(3))/(3) + 3x^(4) + C`.
(iv) `int((3x^(4)-5x^(3)+4x^(2) -x +2)/(x^(3))) dx = int(3x-5+4/x-1/(x^(2)) + 2/(x^(3))) dx`
[dividing each term by `x^(3)`]
`= 3intdx - 5 xdx + 4int1/x dx - intx^(-2) dx + 2 intx^(-3) dx`
`= 3.(x^(2))/(2) - 5x + 4 log |x| - (-1/x) + 2((x^(-2))/(-2)) + C`
`= (3x^(2))/(2) - 5x+4 log |x| + 1/x - 1/(x^(2)) + C`.
(v) `int(x^(2) + 1/(x^(2)))^(3) dx = int(x^(6) + 1/(x^(6)) + 3x^(2) + 3/(x^(2))) dx`
`= intx^(4)dx + intx^(-6)dx + 3 intx^(2) dx + 3int(1)/(x^(2)) dx`
`= (x^(7))/(7) + (x^(-5))/((-5)) + 3.(x^(3))/(3) + 3.(-1/x) + C`.
`= (x^(7))/(7)- (1)/(5x^(5)) + x^(3) - 3/x + C`.