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Engineering Handbook
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Mathematics
Contents
1
Integral
1.1
Indefinite Integral
1.2
Definite Integral
2
References
Integral
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Indefinite Integral
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∫
f
(
x
)
d
x
{\displaystyle \int f(x)dx}
Table of Properties of Integrals
Rule
Conditions
1
∫
a
d
x
=
a
x
{\displaystyle \int a\,dx=ax}
2
Homogeniety
∫
a
f
(
x
)
d
x
=
a
∫
f
(
x
)
d
x
{\displaystyle \int af(x)\,dx=a\int f(x)\,dx}
3
Associativity
∫
(
f
±
g
±
h
±
⋯
)
d
x
=
∫
f
d
x
±
∫
g
d
x
±
∫
h
d
x
±
⋯
{\displaystyle \int {\left(f\pm g\pm h\pm \cdots \right)\,dx}=\int f\,dx\pm \int g\,dx\pm \int h\,dx\pm \cdots }
4
Integration by Parts
∫
a
b
f
g
′
d
x
=
[
f
g
]
a
b
−
∫
a
b
g
f
′
d
x
{\displaystyle \int _{a}^{b}fg'\,dx=\left[fg\right]_{a}^{b}-\int _{a}^{b}gf'\,dx}
4
General Integration by Parts
∫
f
(
n
)
g
d
x
=
f
(
n
−
1
)
g
′
−
f
(
n
−
2
)
g
″
+
…
+
(
−
1
)
n
∫
f
g
(
n
)
d
x
{\displaystyle \int f^{(n)}g\,dx=f^{(n-1)}g'-f^{(n-2)}g''+\ldots +(-1)^{n}\int fg^{(n)}\,dx}
5
∫
f
(
a
x
)
d
x
=
1
a
∫
f
(
x
)
d
x
{\displaystyle \int f(ax)\,dx={\frac {1}{a}}\int f(x)\,dx}
6
Substitution Rule
∫
g
{
f
(
x
)
}
d
x
=
∫
g
(
u
)
d
x
d
u
d
u
=
∫
g
(
u
)
f
′
(
x
)
d
u
{\displaystyle \int g\{f(x)\}\,dx=\int g(u){\frac {dx}{du}}\,du=\int {\frac {g(u)}{f'(x)}}\,du}
u
=
f
(
x
)
{\displaystyle u=f(x)\,}
7
∫
x
n
d
x
=
x
n
+
1
n
+
1
{\displaystyle \int x^{n}\,dx={\frac {x^{n+1}}{n+1}}}
n
≠
−
1
{\displaystyle n\neq -1\,}
8
∫
1
x
d
x
=
ln
|
x
|
{\displaystyle \int {\frac {1}{x}}\,dx=\ln |x|}
9
∫
e
x
d
x
=
e
x
{\displaystyle \int e^{x}\,dx=e^{x}}
10
∫
a
x
d
x
=
a
x
ln
|
a
|
{\displaystyle \int a^{x}\,dx={\frac {a^{x}}{\ln |a|}}}
a
≠
1
{\displaystyle a\neq 1}
Notes:
f, g, h
are functions of
x
a, n
are constants.
The constant of integration,
C
has been omitted from this table. It should be included in the working of the equation if applicable.
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Integral
Value
Remarks
1
∫
c
d
x
{\displaystyle \int c\,dx}
c
x
+
C
{\displaystyle cx+C\,}
2
∫
x
n
d
x
{\displaystyle \int x^{n}\,dx}
x
n
+
1
n
+
1
+
C
{\displaystyle {\frac {x^{n+1}}{n+1}}+C}
n
≠
−
1
{\displaystyle n\neq -1}
3
∫
1
x
d
x
{\displaystyle \int {\frac {1}{x}}\,dx}
ln
|
x
|
+
C
{\displaystyle \ln {\left|x\right|}+C}
4
∫
1
a
2
+
x
2
d
x
{\displaystyle \int {1 \over {a^{2}+x^{2}}}\,dx}
1
a
arctan
x
a
+
C
{\displaystyle {1 \over a}\arctan {x \over a}+C}
5
∫
1
a
2
−
x
2
d
x
{\displaystyle \int {1 \over {\sqrt {a^{2}-x^{2}}}}\,dx}
arcsin
x
a
+
C
{\displaystyle \arcsin {x \over a}+C}
6
∫
−
1
a
2
−
x
2
d
x
{\displaystyle \int {-1 \over {\sqrt {a^{2}-x^{2}}}}\,dx}
arccos
x
a
+
C
{\displaystyle \arccos {x \over a}+C}
7
∫
1
x
x
2
−
a
2
d
x
{\displaystyle \int {1 \over x{\sqrt {x^{2}-a^{2}}}}\,dx}
1
a
arcsec
|
x
|
a
+
C
{\displaystyle {1 \over a}{\mbox{arcsec}}\,{|x| \over a}+C}
8
∫
ln
x
d
x
{\displaystyle \int \ln {x}\,dx}
x
ln
x
−
x
+
C
{\displaystyle x\ln {x}-x+C\,}
9
∫
log
b
x
d
x
{\displaystyle \int \log _{b}{x}\,dx}
x
log
b
x
−
x
log
b
e
+
C
{\displaystyle x\log _{b}{x}-x\log _{b}{e}+C\,}
10
∫
e
x
d
x
{\displaystyle \int e^{x}\,dx}
e
x
+
C
{\displaystyle e^{x}+C\,}
11
∫
a
x
d
x
{\displaystyle \int a^{x}\,dx}
a
x
ln
a
+
C
{\displaystyle {\frac {a^{x}}{\ln {a}}}+C}
12
∫
sin
x
d
x
{\displaystyle \int \sin {x}\,dx}
−
cos
x
+
C
{\displaystyle -\cos {x}+C\,}
13
∫
cos
x
d
x
{\displaystyle \int \cos {x}\,dx}
sin
x
+
C
{\displaystyle \sin {x}+C\,}
14
∫
tan
x
d
x
{\displaystyle \int \tan {x}\,dx}
−
ln
|
cos
x
|
+
C
{\displaystyle -\ln {\left|\cos {x}\right|}+C\,}
15
∫
cot
x
d
x
{\displaystyle \int \cot {x}\,dx}
ln
|
sin
x
|
+
C
{\displaystyle \ln {\left|\sin {x}\right|}+C\,}
16
∫
sec
x
d
x
{\displaystyle \int \sec {x}\,dx}
ln
|
sec
x
+
tan
x
|
+
C
{\displaystyle \ln {\left|\sec {x}+\tan {x}\right|}+C\,}
17
∫
csc
x
d
x
{\displaystyle \int \csc {x}\,dx}
−
ln
|
csc
x
+
cot
x
|
+
C
{\displaystyle -\ln {\left|\csc {x}+\cot {x}\right|}+C\,}
18
∫
sec
2
x
d
x
{\displaystyle \int \sec ^{2}x\,dx}
tan
x
+
C
{\displaystyle \tan x+C\,}
19
∫
csc
2
x
d
x
{\displaystyle \int \csc ^{2}x\,dx}
−
cot
x
+
C
{\displaystyle -\cot x+C\,}
20
∫
sec
x
tan
x
d
x
{\displaystyle \int \sec {x}\,\tan {x}\,dx}
sec
x
+
C
{\displaystyle \sec {x}+C\,}
21
∫
csc
x
cot
x
d
x
{\displaystyle \int \csc {x}\,\cot {x}\,dx}
−
csc
x
+
C
{\displaystyle -\csc {x}+C\,}
22
∫
sin
2
x
d
x
{\displaystyle \int \sin ^{2}x\,dx}
1
2
(
x
−
sin
x
cos
x
)
+
C
{\displaystyle {\frac {1}{2}}(x-\sin x\cos x)+C\,}
23
∫
cos
2
x
d
x
{\displaystyle \int \cos ^{2}x\,dx}
1
2
(
x
+
sin
x
cos
x
)
+
C
{\displaystyle {\frac {1}{2}}(x+\sin x\cos x)+C\,}
24
∫
sin
n
x
d
x
{\displaystyle \int \sin ^{n}x\,dx}
−
sin
n
−
1
x
cos
x
n
+
n
−
1
n
∫
sin
n
−
2
x
d
x
{\displaystyle -{\frac {\sin ^{n-1}{x}\cos {x}}{n}}+{\frac {n-1}{n}}\int \sin ^{n-2}{x}\,dx}
25
∫
cos
n
x
d
x
{\displaystyle \int \cos ^{n}x\,dx}
−
cos
n
−
1
x
sin
x
n
+
n
−
1
n
∫
cos
n
−
2
x
d
x
{\displaystyle -{\frac {\cos ^{n-1}{x}\sin {x}}{n}}+{\frac {n-1}{n}}\int \cos ^{n-2}{x}\,dx}
26
∫
arctan
x
d
x
{\displaystyle \int \arctan {x}\,dx}
x
arctan
x
−
1
2
ln
|
1
+
x
2
|
+
C
{\displaystyle x\,\arctan {x}-{\frac {1}{2}}\ln {\left|1+x^{2}\right|}+C}
27
∫
sinh
x
d
x
{\displaystyle \int \sinh x\,dx}
cosh
x
+
C
{\displaystyle \cosh x+C\,}
28
∫
cosh
x
d
x
{\displaystyle \int \cosh x\,dx}
sinh
x
+
C
{\displaystyle \sinh x+C\,}
29
∫
tanh
x
d
x
{\displaystyle \int \tanh x\,dx}
ln
|
cosh
x
|
+
C
{\displaystyle \ln |\cosh x|+C\,}
30
∫
csch
x
d
x
{\displaystyle \int {\mbox{csch}}\,x\,dx}
ln
|
tanh
x
2
|
+
C
{\displaystyle \ln \left|\tanh {x \over 2}\right|+C}
31
∫
sech
x
d
x
{\displaystyle \int {\mbox{sech}}\,x\,dx}
arctan
(
sinh
x
)
+
C
{\displaystyle \arctan(\sinh x)+C\,}
32
∫
coth
x
d
x
{\displaystyle \int \coth x\,dx}
ln
|
sinh
x
|
+
C
{\displaystyle \ln |\sinh x|+C\,}
Definite Integral
edit
∫
a
b
f
(
x
)
,
d
x
{\displaystyle \int _{a}^{b}f(x),dx}
∫
a
b
f
(
x
)
d
x
=
F
(
b
)
−
F
(
a
)
.
{\displaystyle \int _{a}^{b}f(x)\,dx=F(b)-F(a).}
Furthermore, for every
x
in the interval (
a
,
b
),
d
d
x
∫
a
x
f
(
t
)
d
t
=
f
(
x
)
.
{\displaystyle {\frac {d}{dx}}\int _{a}^{x}f(t)\,dt=f(x).}
References
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Integral
![{\displaystyle \int _{a}^{b}fg'\,dx=\left[fg\right]_{a}^{b}-\int _{a}^{b}gf'\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3990aae3a04d34ec06c70d5e6a3ab6214fd7033)
