From Wikibooks, open books for an open world
Indefinite Integral is a mathematic operation on a non linear function over an indefinite a boundary
∫
f
(
x
)
d
x
=
L
i
m
Δ
x
→
0
Σ
Δ
x
[
f
(
x
)
+
f
(
x
+
Δ
x
)
2
]
{\displaystyle \int f(x)dx=Lim_{\Delta x\to 0}\Sigma \Delta x[f(x)+{\frac {f(x+\Delta x)}{2}}]}
Because antidifferentiation is the inverse operation of the differentiation, antidifferentiation theorems and rules are obtained from those on differentiation. Thus, the following theorems can be proven from the corresponding differentiation theorems:
General antidifferentiation rule:
∫
d
x
=
x
+
C
{\displaystyle \int dx=x+C}
The general antiderivative of a constant times a function is the constant multiplied by the general antiderivative of the function:
∫
a
f
(
x
)
d
x
=
a
∫
f
(
x
)
d
x
+
C
{\displaystyle \int af(x)\,dx=a\int f(x)\,dx+C}
If ƒ and g are defined on the same interval, then the general antiderivative of the sum of ƒ and g equals the sum of the general antiderivatives of ƒ and g :
∫
[
f
(
x
)
+
g
(
x
)
]
d
x
=
∫
f
(
x
)
d
x
+
∫
g
(
x
)
d
x
+
C
{\displaystyle \int [f(x)+g(x)]\,dx=\int f(x)\,dx+\int g(x)\,dx+C}
∫
x
n
d
x
=
{
x
n
+
1
n
+
1
+
C
,
if
n
≠
−
1
ln
|
x
|
+
C
,
if
n
=
−
1
{\displaystyle \int x^{n}\,dx={\begin{cases}{\frac {x^{n+1}}{n+1}}+C,&{\text{if }}n\neq -1\\[6pt]\ln |x|+C,&{\text{if }}n=-1\end{cases}}}
∫
f
(
x
)
d
x
=
f
′
(
x
)
+
C
{\displaystyle \int f(x)dx=f^{'}(x)+C}
∫
f
′
(
x
)
f
(
x
)
d
x
=
ln
|
f
(
x
)
|
+
c
{\displaystyle \int {\frac {f^{'}(x)}{f(x)}}{\rm {d}}x=\ln |f(x)|+c}
∫
U
V
=
U
∫
V
−
∫
(
U
′
∫
V
)
{\displaystyle \int {UV}=U\int {V}-\int {\left(U^{'}\int {V}\right)}}
e
x
{\displaystyle e^{x}}
∫
e
−
x
sin
x
d
x
=
(
−
e
−
x
)
sin
x
−
∫
(
−
e
−
x
)
cos
x
d
x
{\displaystyle \int {e^{-x}\sin x}~dx=(-e^{-x})\sin x-\int {(-e^{-x})\cos x}~dx}
=
−
e
−
x
sin
x
+
∫
e
−
x
cos
x
d
x
{\displaystyle =-e^{-x}\sin x+\int {e^{-x}\cos x}~dx}
=
−
e
−
x
(
sin
x
+
cos
x
)
−
∫
e
−
x
sin
x
d
x
+
c
{\displaystyle =-e^{-x}(\sin x+\cos x)-\int {e^{-x}\sin x}~dx+c}
We now have our required integral on both sides of the equation so
=
−
1
2
e
−
x
(
sin
x
+
cos
x
)
+
c
{\displaystyle =-{\frac {1}{2}}e^{-x}(\sin x+\cos x)+c}
f
(
x
)
=
m
{\displaystyle f(x)=m}
∫
m
d
x
=
m
x
+
C
{\displaystyle \int mdx=mx+C}
f
(
x
)
=
x
n
{\displaystyle f(x)=x^{n}}
∫
f
(
x
)
d
x
=
1
n
+
1
x
n
+
1
+
c
{\displaystyle \int {f(x)}dx={\frac {1}{n+1}}x^{n+1}+c}
f
(
x
)
=
1
x
{\displaystyle f(x)={\frac {1}{x}}}
∫
1
x
d
x
=
ln
x
{\displaystyle \int {\frac {1}{x}}dx=\ln x}
List of integrals of rational functions List of integrals of irrational functions List of integrals of trigonometric functions List of integrals of inverse trigonometric functions List of integrals of hyperbolic functions List of integrals of inverse hyperbolic functions List of integrals of exponential functions List of integrals of logarithmic functions List of integrals of Gaussian functions 
![{\displaystyle \int f(x)dx=Lim_{\Delta x\to 0}\Sigma \Delta x[f(x)+{\frac {f(x+\Delta x)}{2}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b3e848175ab3297fe57ea89932cd9b0340483e4)
![{\displaystyle \int [f(x)+g(x)]\,dx=\int f(x)\,dx+\int g(x)\,dx+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/359d4dde32b10472ed0b93e1637a0834c12fba9d)
![{\displaystyle \int x^{n}\,dx={\begin{cases}{\frac {x^{n+1}}{n+1}}+C,&{\text{if }}n\neq -1\\[6pt]\ln |x|+C,&{\text{if }}n=-1\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/088fb00bf9177731ee2bd62ead036d210712b447)
