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Let the integrand function be
the product of sine and cosine
in positive integer powers of
different parity, for example

sin
x

cos

2m

2n 1

xdx
...


Find integral:

2

I   cos x  sin xdx
...

3

Let the integrand be the
product of sine and cosine in
even degrees:

sin
x

cos
xdx

...

2
2

Find integral:

I   sin xdx
...

2
4

Let the integrand be the
product of trigonometric
functions:

sin
ax
cos
bx
dx
;


sin
ax
sin
bx
dx
;


cos
ax
cos
bx
dx

...

Answer:

1
1
I   cos 6x  cos10x  C
...

2
The practical application
of such a substitution
requires the following
replacements in the
original integral:

x
2tg
2
sin x 
2 x
1  tg
2

2t
sin x 
2
1 t

x
1  tg
2
cos x 
2 x
1  tg
2
2

1 t
cos x 
2
1 t
2

x  2arctgt
2dt
dx 
2
1 t

Prove the table integral:

dx
x

ln
|
tg
|

C

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