Even powers (half-angle identity) - University of Washington

Integrating Powers of Trig

1 cos(ax) dx = sin(ax) + C

a 1 sin(ax) dx = - cos(ax) + C a Even powers (half-angle identity):

cos2(x) dx = 1

11 1 + cos(2x) dx = x + sin(2x) + C

2

24

sin2(x) dx = 1

11 1 - cos(2x) dx = x - sin(2x) + C

2

24

cos4(x) dx

=

1 [ (1

+

cos(2x))]2

dx

=

1

1 + 2 cos(2x) + cos2(2x) dx

2

4

11

1

= x + sin(2x) +

cos2(2x) dx

44

4

11

1

= x + sin(2x) + 1 + cos(4x) dx

44

8

11

11

= x + sin(2x) + x + sin(4x) + C

44

8 32

31

1

= x + sin(2x) + sin(4x) + C

84

32

You can do sin4(x) and sin2(x) cos2(x) is a similar way as above.

Odd powers (identity then substitution):

cos3(x) dx = cos2(x) cos(x) dx = (1 - sin2(x)) cos(x) dx

Here is another example

then use u = sin(x) to get 1 - u2du = u - 1 u3 + C 3 so

cos3(x) dx = sin(x) - 1 sin3(x) + C 3

cos2(x) sin3(x) dx = cos2(x) sin2(x) sin(x) dx = cos2(x)(1 - cos2(x)) sin(x) dx

then use u = cos(x) to get

-u2(1 - u2)du = - 1 u3 + 1 u5 + C 35

so

cos2(x)

sin3(x)

dx

=

1 -

cos3(x)

+

1

cos5(x)

+

C

3

5

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