Trigonometric Integrals
edit
There are a few different types of integrals using trigonometric functions. I will split it into a few different sections. Those involving sine, cosine, and tangent. From there I will cover cotangent, secant, and cosecant. Then I will cover the inverse functions, functions involving e, ln, and finally hyperbolic functions.
Something to keep in mind is that the variable used in these functions are denoted by u . (see substitution section)
sin
u
{\displaystyle \sin u}
cos
u
{\displaystyle \cos u}
tan
u
{\displaystyle \tan u}
edit
∫
sin
u
d
u
=
−
cos
u
+
C
{\displaystyle \int \sin u\mathrm {d} u=-\cos u+C}
∫
cos
u
d
u
=
sin
u
+
C
{\displaystyle \int \cos u\mathrm {d} u=\sin u+C}
∫
tan
u
d
u
=
−
ln
|
cos
u
|
+
C
{\displaystyle \int \tan u\mathrm {d} u=-\ln \left\vert \cos u\right\vert +C}
∫
sin
2
u
d
u
=
1
2
(
u
−
sin
u
cos
u
)
+
C
{\displaystyle \int \sin ^{2}u\mathrm {d} u={\frac {1}{2}}(u-\sin u\cos u)+C}
∫
cos
2
u
d
u
=
1
2
(
u
+
sin
u
cos
u
)
+
C
{\displaystyle \int \cos ^{2}u\mathrm {d} u={\frac {1}{2}}(u+\sin u\cos u)+C}
∫
tan
2
u
d
u
=
−
u
+
tan
u
+
C
{\displaystyle \int \tan ^{2}u\mathrm {d} u=-u+\tan u+C}
∫
sin
k
u
d
u
=
−
sin
k
−
1
u
cos
u
k
+
k
−
1
k
∫
sin
k
−
2
u
d
u
{\displaystyle \int \sin ^{k}u\mathrm {d} u=-{\frac {\sin ^{k-1}u\cos u}{k}}+{\frac {k-1}{k}}\int \sin ^{k-2}u\mathrm {d} u}
∫
cos
k
u
d
u
=
cos
k
−
1
u
sin
u
k
+
k
−
1
k
∫
cos
k
−
2
u
d
u
{\displaystyle \int \cos ^{k}u\mathrm {d} u={\frac {\cos ^{k-1}u\sin u}{k}}+{\frac {k-1}{k}}\int \cos ^{k-2}u\mathrm {d} u}
∫
tan
k
u
d
u
=
cos
k
−
1
u
sin
u
k
+
k
−
1
k
∫
cos
k
−
2
u
d
u
{\displaystyle \int \tan ^{k}u\mathrm {d} u={\frac {\cos ^{k-1}u\sin u}{k}}+{\frac {k-1}{k}}\int \cos ^{k-2}u\mathrm {d} u}
∫
u
sin
u
d
u
=
sin
u
−
u
cos
u
+
C
{\displaystyle \int u\sin u\mathrm {d} u=\sin u-u\cos u+C}
∫
u
cos
u
d
u
=
cos
u
+
u
sin
u
+
C
{\displaystyle \int u\cos u\mathrm {d} u=\cos u+u\sin u+C}
∫
u
k
sin
u
d
u
=
−
u
k
cos
u
+
k
∫
u
k
−
1
cos
u
d
u
{\displaystyle \int u^{k}\sin u\mathrm {d} u=-u^{k}\cos u+k\int u^{k-1}\cos u\mathrm {d} u}
∫
u
k
cos
u
d
u
=
u
k
sin
u
−
k
∫
u
k
−
1
sin
u
d
u
{\displaystyle \int u^{k}\cos u\mathrm {d} u=u^{k}\sin u-k\int u^{k-1}\sin u\mathrm {d} u}
∫
1
1
±
sin
u
d
u
=
tan
u
±
sec
u
+
C
{\displaystyle \int {\frac {1}{1\pm \sin u}}\mathrm {d} u=\tan u\pm \sec u+C}
∫
1
1
±
cos
u
d
u
=
−
cot
u
±
csc
u
+
C
{\displaystyle \int {\frac {1}{1\pm \cos u}}\mathrm {d} u=-\cot u\pm \csc u+C}
∫
1
1
±
tan
u
d
u
=
1
2
(
u
±
ln
|
cos
u
±
sin
u
|
)
+
C
{\displaystyle \int {\frac {1}{1\pm \tan u}}\mathrm {d} u={\frac {1}{2}}(u\pm \ln \left\vert \cos u\pm \sin u\right\vert )+C}
∫
1
sin
u
cos
u
d
u
=
ln
|
tan
u
|
+
C
{\displaystyle \int {\frac {1}{\sin u\cos u}}\mathrm {d} u=\ln \left\vert \tan u\right\vert +C}
cot
u
{\displaystyle \cot u}
sec
u
{\displaystyle \sec u}
csc
u
{\displaystyle \csc u}
edit
∫
cot
u
d
u
=
ln
|
sin
u
|
+
C
{\displaystyle \int \cot u\mathrm {d} u=\ln \left\vert \sin u\right\vert +C}
∫
sec
u
d
u
=
ln
|
sec
u
+
tan
u
|
+
C
{\displaystyle \int \sec u\mathrm {d} u=\ln \left\vert \sec u+\tan u\right\vert +C}
∫
csc
u
d
u
=
ln
|
csc
u
−
cot
u
|
+
C
{\displaystyle \int \csc u\mathrm {d} u=\ln \left\vert \csc u-\cot u\right\vert +C}
∫
cot
2
u
d
u
=
−
u
−
cot
u
+
C
{\displaystyle \int \cot ^{2}u\mathrm {d} u=-u-\cot u+C}
∫
sec
2
u
d
u
=
tan
u
+
C
{\displaystyle \int \sec ^{2}u\mathrm {d} u=\tan u+C}
∫
csc
2
u
d
u
=
−
cot
u
+
C
{\displaystyle \int \csc ^{2}u\mathrm {d} u=-\cot u+C}
∫
cot
k
u
d
u
=
−
cot
k
−
1
u
k
−
1
∫
cot
k
−
2
u
d
,
k
≠
1
{\displaystyle \int \cot ^{k}u\mathrm {d} u=-{\frac {\cot ^{k-1}u}{k-1}}\int \cot ^{k-2}u\mathrm {d} ,k\neq 1}
∫
sec
k
u
d
u
=
sec
k
−
2
u
tan
u
k
−
1
+
k
−
2
k
−
1
∫
sec
k
−
2
u
d
u
,
l
≠
1
{\displaystyle \int \sec ^{k}u\mathrm {d} u={\frac {\sec ^{k-2}u\tan u}{k-1}}+{\frac {k-2}{k-1}}\int \sec ^{k-2}u\mathrm {d} u,l\neq 1}
∫
csc
k
u
d
u
=
−
csc
k
−
2
u
tan
u
k
−
1
+
k
−
2
k
−
1
∫
csc
k
−
2
u
d
u
,
k
/
n
e
q
1
{\displaystyle \int \csc ^{k}u\mathrm {d} u=-{\frac {\csc ^{k-2}u\tan u}{k-1}}+{\frac {k-2}{k-1}}\int \csc ^{k-2}u\mathrm {d} u,k/neq1}
∫
1
1
±
cot
u
d
u
=
1
2
(
u
∓
ln
|
sin
u
±
cos
u
|
)
+
C
{\displaystyle \int {\frac {1}{1\pm \cot u}}\mathrm {d} u={\frac {1}{2}}(u\mp \ln \left\vert \sin u\pm \cos u\right\vert )+C}
∫
1
1
±
sec
u
d
u
=
u
+
cot
u
∓
csc
u
+
C
{\displaystyle \int {\frac {1}{1\pm \sec u}}\mathrm {d} u=u+\cot u\mp \csc u+C}
∫
1
1
±
csc
d
u
=
u
−
tan
u
±
sec
u
+
C
{\displaystyle \int {\frac {1}{1\pm \csc }}\mathrm {d} u=u-\tan u\pm \sec u+C}
Inverse Trig Functions
edit
∫
arcsin
u
d
u
=
u
arcsin
u
+
1
−
u
2
+
C
{\displaystyle \int \arcsin u\mathrm {d} u=u\arcsin u+{\sqrt {1-u^{2}}}+C}
∫
arccos
u
d
u
=
u
arccos
u
−
1
−
u
2
+
C
{\displaystyle \int \arccos u\mathrm {d} u=u\arccos u-{\sqrt {1-u^{2}}}+C}
∫
arctan
u
d
u
=
u
arctan
u
−
ln
1
+
u
2
+
C
{\displaystyle \int \arctan u\mathrm {d} u=u\arctan u-\ln {\sqrt {1+u^{2}}}+C}
∫
arccot
u
d
u
=
u
arccot
u
+
ln
1
+
u
2
+
C
{\displaystyle \int \operatorname {arccot} u\mathrm {d} u=u\operatorname {arccot} u+\ln {\sqrt {1+u^{2}}}+C}
∫
arcsec
u
d
u
=
u
arcsec
u
+
ln
|
u
+
u
2
−
1
|
+
C
{\displaystyle \int \operatorname {arcsec} u\mathrm {d} u=u\operatorname {arcsec} u+\ln \left\vert u+{\sqrt {u^{2}-1}}\right\vert +C}
∫
arccsc
u
d
u
=
u
arccsc
u
+
ln
|
u
+
u
2
−
1
|
+
C
{\displaystyle \int \operatorname {arccsc} u\mathrm {d} u=u\operatorname {arccsc} u+\ln \left\vert u+{\sqrt {u^{2}-1}}\right\vert +C}
e
u
{\displaystyle e^{u}}
edit
∫
e
u
d
u
=
e
u
+
C
{\displaystyle \int e^{u}\mathrm {d} u=e^{u}+C}
∫
u
e
u
d
u
=
(
u
−
1
)
e
u
+
C
{\displaystyle \int ue^{u}\mathrm {d} u=(u-1)e^{u}+C}
∫
u
k
e
u
d
u
=
k
∫
u
k
−
1
e
u
d
u
{\displaystyle \int u^{k}e^{u}\mathrm {d} u=k\int u^{k-1}e^{u}\mathrm {d} u}
∫
1
1
+
e
u
d
u
−
u
−
ln
(
1
+
e
u
)
+
C
{\displaystyle \int {\frac {1}{1+e^{u}}}\mathrm {d} u-u-\ln(1+e^{u})+C}
∫
e
a
u
sin
b
u
d
u
=
e
a
u
a
2
+
b
2
(
a
sin
b
u
−
b
cos
b
u
)
+
C
{\displaystyle \int e^{au}\sin bu\mathrm {d} u={\frac {e^{au}}{a^{2}+b^{2}}}(a\sin bu-b\cos bu)+C}
∫
e
a
u
cos
b
u
d
u
=
e
a
u
a
2
+
b
2
(
a
cos
b
u
+
b
sin
b
u
)
+
C
{\displaystyle \int e^{au}\cos bu\mathrm {d} u={\frac {e^{au}}{a^{2}+b^{2}}}(a\cos bu+b\sin bu)+C}
ln
u
{\displaystyle \ln u}
edit
∫
]
l
n
u
d
u
=
u
(
−
1
+
ln
u
)
+
C
{\displaystyle \int ]lnu\mathrm {d} u=u(-1+\ln u)+C}
∫
u
ln
u
d
u
=
u
2
4
(
−
1
+
2
ln
u
)
+
C
{\displaystyle \int u\ln u\mathrm {d} u={\frac {u^{2}}{4}}(-1+2\ln u)+C}
∫
u
k
ln
u
d
u
=
e
a
u
a
2
+
b
2
(
a
cos
b
u
+
b
sin
b
u
)
+
C
{\displaystyle \int u^{k}\ln u\mathrm {d} u={\frac {e^{au}}{a^{2}+b^{2}}}(a\cos bu+b\sin bu)+C}
∫
(
ln
u
)
2
d
u
=
u
[
2
−
2
ln
u
+
(
ln
u
)
2
]
+
C
{\displaystyle \int (\ln u)^{2}\mathrm {d} u=u[2-2\ln u+(\ln u)^{2}]+C}
∫
(
ln
u
)
k
d
u
=
u
(
ln
u
)
k
−
k
∫
(
ln
u
)
k
−
1
d
u
{\displaystyle \int (\ln u)^{k}\mathrm {d} u=u(\ln u)^{k}-k\int (\ln u)^{k-1}\mathrm {d} u}
Hyperbolic Functions
edit
∫
cosh
u
d
u
=
sinh
u
+
C
{\displaystyle \int \cosh u\mathrm {d} u=\sinh u+C}
∫
sinh
u
d
u
=
cosh
u
+
C
{\displaystyle \int \sinh u\mathrm {d} u=\cosh u+C}
∫
sech
2
u
d
u
=
tanh
u
+
C
{\displaystyle \int \operatorname {sech} ^{2}u\mathrm {d} u=\tanh u+C}
∫
csch
2
u
d
u
=
−
coth
u
+
C
{\displaystyle \int \operatorname {csch} ^{2}u\mathrm {d} u=-\coth u+C}
∫
sech
u
tan
u
d
u
=
−
sech
u
+
C
{\displaystyle \int \operatorname {sech} u\tan u\mathrm {d} u=-\operatorname {sech} u+C}
∫
csch
u
coth
u
d
u
=
−
csch
u
+
C
{\displaystyle \int \operatorname {csch} u\coth u\mathrm {d} u=-\operatorname {csch} u+C}