(1.)  xe3x dx            here is the problem

Let u = x           Let dv = e3x

du = dx             v = (1/3)e3x

uv -
v du           use the parts formula

= (1/3)xe3x - (1/3)
e3x dx    make substitutions

=  (1/3)xe3x - (1/3)(1/3)e3x + C     integrate

= (1/3)xe3x - (1/9)e3x + C       multiply

(2.)
xe-7x dx             here is the problem

Let u = x             Let dv = e-7x

du = dx                v = (-1/7)e-7x

uv -
v du           use the parts formula

=  (-1/7)xe-7x + (1/7)
e-7x dx    make substitutions

=  (-1/7)xe-7x + (1/7)(-1/7)e-7x + C     integrate

=  (-1/7)xe-7x - (1/49)e-7x + C     multiply

(3.)
x2e-x dx               here is the problem

Let u = x2             Let dv = e-x

du = 2x dx        v = -e-x

uv -
v du           use the parts formula

=  -x2e-x + 2
xe-x dx     make substitutions

Let u = x      Let dv = e-x

du = dx        v = -e-x

=  -x2e-x + 2[uv -
v du]   use the parts formula

=  -x2e-x - 2xe-x + 2
e-x dx    make substitutions

=  -x2e-x - 2xe-x - 2e-x + C    integrate

(4.)
x2ex/4 dx

(5.)
x ln x dx            here is the problem

Let u = ln x               Let dv = x

du = (1/x)dx             v = (1/2)x2

uv -
v du           use the parts formula

=  (1/2)x2 ln x - (1/2)
x2 * (1/x)dx    make substitutions

= (1/2)x2 ln x - (1/2)
x dx      cancel

= (1/2)x2 ln x - (1/2)(1/2)x2 + C    integrate

= (1/2)x2 ln x - (1/4)x2 + C            multiply

(6.)
x7 ln x dx              here is the problem

Let u = ln x                      Let dv = x7

du = (1/x)dx                v = (1/8)x8

uv -
v du             use the parts formula

=  (1/8)x8 ln x - (1/8)
(x8)(1/x)dx   make substitutions

=  (1/8)x8 ln x - (1/8)
x7 dx        cancel

= (1/8)x8 ln x - (1/8)(1/8)x8 + C    integrate

=  (1/8)x8 ln x - (1/64)x8 + C     multiply

(7.)
x sinh x dx          here is the problem

let u = x             Let dv = sinh x

du = dx                 v = cosh x

uv -
v du              use the parts formula

=   x cosh x -
cosh x dx    make substitutions

=   x cosh x - sinh x + C    integrate

(8.)
x sin x dx           here is the problem

let u = x                Let dv = sin x

du = dx                  v = -cos x

uv -
v du                     use the parts formula

=   -x cos x +
cos x dx          make substitutions

=   -x cos x + sin x + C           integrate

(9.)
x[1 - (x/2)]1/2 dx

Let u = x                    Let dv = [1 - (x/2)]1/2

du = dx                  v = -(4/3)[1 - (x/2)]3/2

uv -
v du                 use the parts formula

=  (-4/3)x[1 - (x/2)]3/2 + (4/3)
[1 - (x/2)]3/2 dx

[make substitutions]

=  (-4/3)x[1 - (x/2)]3/2 + (4/3)(-2)(2/5)[1 - (x/2)]5/2 + C

[integrate]

=  (-4/3)x[1 - (x/2)]3/2 - (16/15)[1 - (x/2)]5/2 + C  multiply

(10.)
x(3x + 1)1/2 dx

Let u = x

du = dx

Let dv = (3x + 1)1/2

v = (2/9)(3x + 1)3/2

uv -
v du              use the parts formula

=   (2x/9)(3x + 1)3/2 - (2/9)
(3x + 1)3/2 dx  make substitutions

=  (2x/9)(3x + 1)3/2 - (2/9)(1/3)(2/5)(3x + 1)5/2 + C  integrate

=  (2x/9)(3x + 1)3/2 - (4/135)(3x + 1)5/2 + C     multiply

(11.)
x2 cosh 2x dx

Let u = x2

du = 2x dx

Let dv = cosh 2x

v = (1/2)sinh 2x

uv -
v du                 use the parts formula

=   (1/2)x2 sinh 2x -
x sinh 2x dx   make substitutions

Let u = x

du = dx

Let dv = sinh 2x

v = (1/2)cosh 2x

=  (1/2)x2 sinh 2x - [uv -
v du]   use parts again

=  (1/2)x2 sinh 2x - (1/2)x + (1/2)
cosh 2x dx

[make substitutions]

= (1/2)x2 sinh 2x - (1/2)x + (1/2)(1/2)sinh 2x + C  integrate

= (1/2)x2 sinh 2x - (1/2)x + (1/4)sinh 2x + C     multiply

(12.)
x2 cos 2x dx

(13.)
cos (ln x) dx

Let u = ln x

x = eu

dx = eu du

cos u * eu du      make substitutions

let w = cos u

dw = -sin u du

Let dv = eu

v = eu

wv -
v dw       use the parts formula

=   eu cos u +
eu sin u du     make substitutions

Let w = sin u

dw = cos u du

Let dv = eu

v = eu

=   eu cos u + [wv -
v dw]

=   eu cos u + eu sin u -
eu cos u du     make substitutions
2
eu cos u du = eu cos u + eu sin u

eu cos u du = (1/2)eu cos u + (1/2) eu sin u

elnx cos (ln x)(1/x)   replace u with ln x and du with 1/x

cos (ln x)                cancel

=   (1/2)x cos (ln x) + (1/2) x sin (ln x)   make substitutions

(14.)
(ln x)2 dx

Let u = (ln x)2

du = (2/x)(ln x)dx

Let dv = dx

v = x

uv -
v du         use the parts formula

=   x(ln x)2 - 2
(ln x)dx       make substitutions

Let u = ln x

du = dx/x

Let dv = dx

v = x

= x(ln x)2 - 2[uv -
v du]     use the parts formula again

=  x(ln x)2 - 2x ln x + 2
dx   make substitutions

=  x(ln x)2 - 2x ln x + 2x + C   integrate

(15.)
x5 ex^3 dx