(1.)  [x/(x - 1)4]dx             here is the problem

Let u = x - 1

du = dx

x = u + 1

(u + 1)du/u4     make substitutions

=
(u-3 + u-4)du          divide thru by u4

=   (-1/2)u-2 - (1/3)u-3 + C    integrate

=  (-1/2)(x - 1)-2 - (1/3)(x - 1)-3 + C  replace u with x - 1

-1                 1
=   ____________ -    __________ + C   use positive exponents
2(x - 1)2         3(x - 1)3

(2.)
dx/(4 + 5x1/2)            here is the problem

let u = 5x1/2

u2 = 25x           square each side

2u du = 25 dx         take the derivative of each side

(2/25)u du = dx         multiply by 1/25 each side

(2/25)
(u du)/(4 + u)    make substitutions

=  (2/25)
(u + 4 - 4)du/(4 + u)    add 4 and subtract 4

=  (2/25)
du - (8/25)[du/(4 + u)]   separate like this

=  (2/25)u - (8/25) ln (4 + u) + C   integrate

=  (2/25)(5x1/2) - (8/25) ln (4 + 5x1/2) + C   replace u with 5x1/2

(3.)
x(1 + 2x)2/3 dx

Let u = 1 + 2x

du = 2 dx

(1/2)du = dx

(1/2)u - (1/2) = x

[(1/2)u - (1/2)](u2/3)[(1/2)du]  make substitutions

=  (1/4)
u5/3 du - (1/4)u2/3 du    multiply thru

=  (1/4)(3/8)u8/3 - (1/4)(3/5)u5/3 + C  integrate

=  (3/32)u8/3 - (3/20)u5/3 + C        multiply

=  (3/32)(1 + 2x)8/3 - (3/20)(1 + 2x)5/3 + C   make substitutions

(4.)
(3x5)/(1 + x3)3/2  dx          here is the problem

Let u = 1 + x3

x3 = u - 1

x = (u - 1)1/3

dx = (1/3)(u - 1)-2/3 du

3
∫[(u - 1)5/3 * (1/3)(u - 1)-2/3]/u3/2   make substitutions

=
(u - 1)/u3/2   du        simplify, add exponents

=
[u-1/2 - u-3/2]du       divide

=    2u1/2 + 2u-1/2  +  C         integrate

=   2(1 + x3)1/2 + 2(1 + x3)-1/2 + C       make substitutions

(5.)
(x8)(1 - x3)1/3 dx                here is the problem

Let u = 1 - x3

u - 1 = -x3

1 - u = x3

(1 - u)1/3 = x

dx = -(1/3)(1 - u)-2/3 du

(1 - u)8/3 * u1/3 * (-1/3)(1 - u)-2/3 du  make substitutions

=  (-1/3)
(1 - u)2 * u1/3 du    add exponents

=  (-1/3)
(1 - 2u + u2)(u1/3)du       square the binomial

=  (-1/3)
(u1/3 - 2u4/3 + u7/3)du    multiply thru parentheses

=  (-1/3)[(3/4)u4/3 - (6/7)u7/3 + (3/10)u10/3 + C]  integrate

=  (-1/4)u4/3 + (2/7)u7/3 - (1/10)u10/3 + C    multiply thru

=  (-1/4)(1 - x3)4/3 + (2/7)(1 - x3)7/3 - (1/10)(1 - x3)10/3 + C

[make substitutions]

(6.)
(x8)(1 - x3)6/5 dx          here is the problem
Let u = 1 - x3

x3 = 1 - u

x = (1 - u)1/3

dx = (-1/3)(1 - u)-2/3 du

(1 - u)8/3 * u6/5 * (-1/3)(1 - u)-2/3 du  make substitutions

= (-1/3)
(1 - u)2 * u6/5 du         add exponents

= (-1/3)
(1 - 2u + u2)(u6/5)du     square the binomial

= (-1/3)
[u6/5 - 2u11/5 + u16/5]du    multiply thru parentheses

= (-1/3)[(5/11)u11/5 - (5/8)u16/5 + (5/21)u21/5] + C  integrate

=   (-5/33)u11/5 + (1/24)u16/5 - (5/63)u21/5 + C    multiply thru

=  (-5/33)(1 - x3)11/5 + (1/24)(1 - x3)16/5 - (5/63)(1 - x3)21/5 + C

[make substitutions]

(7.)
x(1 + x)1/2 dx           here is the problem

Let u = 1 + x

x = u - 1

dx = du

(u - 1)(u)1/2 du         make substitutions

=
[u3/2 - u1/2]du             multiply thru parentheses

=  (2/5)u5/2 - (2/3)u3/2 + C       integrate

=  (2/5((1 + x)5/2 - (2/3)(1 + x)3/2 +  C    make substitutions

(8.)
(1 + x)/(1 - x)1/2   dx          here is the problem

Let u = 1 - x

du = -dx

-du = dx

x = 1 - u

-
(2 - u)/(u1/2) du        make substitutions

=
(u - 2)/(u1/2)  du             simplify

=
[u1/2 - 2u-1/2]du          divide

=   (2/3)u3/2 - 4u1/2 + C   integrate

=  (2/3)(1 - x)3/2 - 4(1 - x)1/2 + C      make substitutions

(9.)
(3x2 - x)/(1 + x)1/2   dx

Let u = 1 + x

du = dx

x = u - 1

[3(u - 1)2 - (u - 1)]du/(u1/2)     make substitutions

=
[(3u2 - 6u + 3) - (u - 1)]du/(u1/2)   square the binomial

=
(2u2 - 7u + 4)du/(u1/2)   combine like terms on top

=
[2u3/2 - 7u1/2 + 4u-1/2]du    divide

=   (4/5)u5/2 - (14/3)u3/2 + 8u1/2 + C    integrate

=  (4/5)(1 + x)5/2 - (14/3)(1 + x)3/2 + 8(1 + x)1/2 + C

[make substitutions]

(10.)
(x3 - x)/(x2 - 2)1/2   dx

Let u = x2 - 2

x2 = u + 2

x = (u + 2)1/2

dx = (1/2)(u + 2)-1/2 du

(1/2)
[(u + 2)3/2 - (u + 2)1/2](u + 2)-1/2 du   make substitutions

= (1/2)
[(u + 2) - 1]du     multiply thru add exponents

= (1/2)
(u + 1)du                combine like terms

=  (1/2)(1/2)u2 + (1/2)u + C   integrate

= (1/4)(x2 - 2)2 + (1/2)(x2 - 2) +  C

[make substitutions & multiply]