Download Trigonometric Substitutions Substitutions • u = atanµ replaces a 2 +

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Trigonometric Substitutions
Substitutions
² u = a tan µ replaces a2 + u2
requires µ = tan¡1( ua ) with
² u = a sin µ
¼
2
<µ<
¼
2
·µ·
¼
2
replaces a2 ¡ u2
requires µ = sin¡1 ( au ) with
² u = a sec µ
¡
¡
¼
2
replaces u2 ¡ a2
requires µ =
sec¡1( ua )
with
8
>
< 0·µ<
>
:
¼
2
¼
2
if
u
a
¸1
< µ · ¼ if
u
a
· ¡1
Useful Trigonometry
sin2 µ + cos2 µ = 1
Integrals of Trigonometric Functions
tan2 µ + 1 = sec2 µ
= ¡ cos µ + C
1 + cot2 µ = csc2 µ
Z
sin µ dµ
Z
cos µ dµ =
Z
Z
sin2 µ =
1 ¡ cos 2µ
2
tan µ dµ = ¡ ln j cos µj + C
cos2 µ =
1 + cos 2µ
2
=
ln j sec µj + C
tan2 µ =
1 ¡ cos 2µ
1 + cos 2µ
cot µ dµ =
ln j sin µj + C
sin 2µ = 2 sin µ cos µ
sin µ + C
= ¡ ln j csc µj + C
Z
sec µ dµ
Z
csc µ dµ
=
ln j sec µ + tan µj + C
= ¡ ln j csc µ + cot µj + C
cos 2µ = cos2 µ ¡ sin2 µ
= 2 cos2 µ ¡ 1
= 1 ¡ 2 sin2 µ
tan 2µ =
2 tan µ
1 ¡ tan2 µ
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