∫ dx
a^2 −b^2 cos^2 x=
1
a
√
a^2 −b^2
tan−^1
( a
√
a^2 −b^2
tanx
)
+C, a > b
− 1
a√b^2 −a^2 tanh
− 1 (√ a
b^2 −a^2 tanx
)
+C, b > a
∫ dx
(acosx+bsinx)^2 =
1
a^2 +b^2 tan
(
x−tan−^1 ab
)
+C
∫ sinx dx
√
acos^2 x+ 2bcosx+c
=
√−^1
−asin
− 1
(√
−a(acos√^2 x+ 2bcosx+c)
b^2 −ac
)
+C, b^2 > ac, a < 0
√−^1
asinh
− 1
(√
a(acos√^2 x+ 2bcosx+c)
b^2 −ac
)
+C, b^2 > ac, a > 0
√−^1
acosh
− 1
(√
a(acos√^2 x+ 2bcosx+c)
ac−b^2
)
+C, b^2 < ac, a > 0
∫ cosx dx
√
asin^2 x+ 2bsinx+c
=
√^1
−asin
− 1
√
−a(asin^2 x+ 2bsinx+c)
√
b^2 −ac
+C, b^2 > ac, a < 0
√^1
asinh
− 1
√
a(asin^2 x+ 2bsinx+c)
√
b^2 −ac
+C, b^2 > ac, a > 0
√^1
acosh
− 1
√
a(asin^2 x+ 2bsinx+c)
√
ac−b^2
+C, b^2 < ac, a > 0
Integrals containing tanax, cotax, secax, cscax
Write integrals in terms ofsinaxandcosaxand see previous listings.
Integrals containing inverse trigonmetric functions
∫
sin−^1 xadx=xsin−^1 xa+
√
a^2 −x^2 +C
∫
cos−^1 xadx=xcos−^1 xa−
√
a^2 −x^2 +C
∫
tan−^1 xadx=xtan−^1 xa−a 2 ln|x^2 +a^2 |+C
∫
cot−^1 xadx=xcot−^1 xa+a 2 ln|x^2 +a^2 |+C
∫
sec−^1 xadx=
xsec−^1 xa−aln|x+
√
x^2 −a^2 |+C, 0 <sec−^1 xa< π/ 2
xsec−^1 xa+aln|x+
√
x^2 −a^2 +C, π/ 2 <sec−^1 xa< π
∫
csc−^1 xadx=
xcsc−^1 xa+aln|x+
√
x^2 −a^2 |+C, 0 <csc−^1 xa< π/ 2
xcsc−^1 xa−aln|x+
√
x^2 −a^2 |+C, −π/ 2 <csc−^1 xa< 0
∫
xsin−^1 xadx=
(x 2
2 −
a^2
4
)
sin−^1 xa+^14 x
√
a^2 −x^2 +C
∫
xcos−^1 xadx=
(x 2
2 −
a^2
4
)
cos−^1 xa−^14 x
√
a^2 −x^2 +C
Appendix C
