(1.)  sin3 x cos2 x dx          here is the problem

=
(cos2 x)(1 - cos2 x)(sin x)dx    factor like this

=
cos2 x sin x dx - cos4 x sin x dx   multiply thru

=    (-1/3)cos3 x + (1/5)cos5 x + C   integrate

(2.)
sin3 x sec x dx                here is the problem

=
tan x sec2 x dx                 use this trig id

=    (1/2)tan2 x + C                  integrate

(3.)
cos3 x sin1/2 x dx             here is the problem

=
(1 - sin2 x)(cos x)(sin1/2 x)dx    factor

=
sin1/2 x cos x dx - sin5/2 x cos x dx   multiply thru

=   (2/3)sin3/2 x - (2/7)sin7/2 x + C     integrate

(4.)
cos3 x csc1/2 x dx            here is the problem

=
cos3 x sin-1/2 x dx       reciprocal identity

=
(1 - sin2 x)(cos x)(sin-1/2 x)dx    trig id, factor

=
sin-1/2 x cos x -  sin3/2 x cos x dx   multiply thru

=   2 sin1/2 x - (2/5)sin5/2 x + C    integrate

(5.)
sin5 x cos2 x dx         here is the problem

=
(1 - cos2 x)2 * (cos2 x)(sin x) dx    trig id, factor

=
(1 - 2 cos2 x + cos4 x)(cos2 x)(sin x)dx  square the binomial

=
cos2 x sin x dx - 2cos4 x sin x + cos6 x sin x dx

[multiply thru parentheses]

=   (-1/3)cos3 x + (2/5)cos5 x - (1/7)cos7 x + C  integrate

(6.)
sin5 x sec2 x dx          here is the problem

=
(sin5 x)(cos-2 x)dx           reciprocal identity

=
(1 - cos2 x)2(sin x)(cos-2 x)dx   trig id

=
(1 - 2cos2 x + cos4 x)(sin x)(cos-2 x)dx

[square the binomial]

=
cos-2 x sin x dx - 2sin x dx + cos2 x sin x dx

[multiply thru parentheses]

=   -(cos x)-1 + 2 cos x - (1/3)cos3 x + C   integrate

=    - sec x + 2 cos x - (1/3)cos3 x + C      reciprocal id

(7.)
sin2 x cos5 x dx          here is the problem

=
(sin2 x)(1 - sin2 x)2(cos x)dx    factor, trig id

=
(sin2 x)(1 - 2sin2 x + sin4 x)(cos x)dx

[square the binomial]

=
sin2 x cos x dx - 2sin4 x cos x dx + sin6 x cos x dx

[multiply thru parentheses]

=  (1/3)sin3 x - (2/5)sin5 x + (1/7)sin7 x + C     integrate

(8.)
sin2 x tan x dx            here is the problem

=
sin3 x sec x dx         trig id

=
(1 - cos2 x)(sin x)(cos x)-1 dx     trig id's

=
(cos x)-1 sin x dx - cos x sin x dx   multiply thru

=  - ln cos x - (1/2)sin2 x + C    integrate

(9.)
cos2 x dx          here is the problem

=    (1/2)
(1 + cos 2x)dx          half angle id for cos

=   (1/2)
dx + (1/2)cos 2x dx      multiply thru parentheses

=  (1/2)x + (1/4)sin 2x + C    integrate

(10.)
sin4 x dx              here is the problem

=   (1/4)
(1 - cos 2x)2 dx      half angle id for sine

=  (1/4)
(1 - 2 cos 2x + cos2 2x) dx   square the binomial

=  (1/4)
dx - (1/2)cos 2x dx + (1/4)(1/2)(1 + cos 4x)dx

[multiply thru parentheses][half angle id for cos]

=  (1/4)
dx - (1/2)cos 2x dx + (1/8)dx + (1/8)cos 4x dx

[multiply thru parentheses]

=  (1/4)x - (1/2)(1/2)sin 2x + (1/8)x + (1/8)(1/4)sin 4x + C

=  (3/8)x - (1/4)sin 2x + (1/32)sin 4x + C    multiply

[and combine like terms]