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How to use it?
- Integral of d{x}:
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Integral of (4x)/(1+x^2)
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Integral of 4^(2*x)
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Integral of 1/(1-exp(y))
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Integral of x/(x^2-3x+3)
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Identical expressions
- x/(x^ two - three x+3)
- x divide by (x squared minus 3x plus 3)
- x divide by (x to the power of two minus three x plus 3)
- x/(x2-3x+3)
- x/x2-3x+3
- x/(x²-3x+3)
- x/(x to the power of 2-3x+3)
- x/x^2-3x+3
- x divide by (x^2-3x+3)
- x/(x^2-3x+3)dx
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Similar expressions
- x/(x^2+3x+3)
- x/(x^2-3x-3)
Integral of x/(x^2-3x+3) dx
The solution
You have entered
[src]
1 / | | x | ------------ dx | 2 | x - 3*x + 3 | / 0
$$\int\limits_{0}^{1} \frac{x}{\left(x^{2} - 3 x\right) + 3}\, dx$$
Integral(x/(x^2 - 3*x + 3), (x, 0, 1))
Detail solution
We have the integral:
/ | | x | ------------ dx | 2 | x - 3*x + 3 | /
Rewrite the integrand
/ 2*x - 3 \
|------------| / 3 \
| 2 | |-----|
x \x - 3*x + 3/ \2*3/4/
------------ = -------------- + -------------------------
2 2 2
x - 3*x + 3 / ___ \
|-2*\/ 3 ___|
|--------*x + \/ 3 | + 1
\ 3 / or
/ | | x | ------------ dx | 2 = | x - 3*x + 3 | /
/
|
| 2*x - 3
| ------------ dx
| 2
| x - 3*x + 3 /
| |
/ | 1
------------------ + 2* | ------------------------- dx
2 | 2
| / ___ \
| |-2*\/ 3 ___|
| |--------*x + \/ 3 | + 1
| \ 3 /
|
/ In the integral
/
|
| 2*x - 3
| ------------ dx
| 2
| x - 3*x + 3
|
/
------------------
2 do replacement
2 u = x - 3*x
then
the integral =
/
|
| 1
| ----- du
| 3 + u
|
/ log(3 + u)
----------- = ----------
2 2 do backward replacement
/
|
| 2*x - 3
| ------------ dx
| 2
| x - 3*x + 3
| / 2 \
/ log\3 + x - 3*x/
------------------ = -----------------
2 2 In the integral
/ | | 1 2* | ------------------------- dx | 2 | / ___ \ | |-2*\/ 3 ___| | |--------*x + \/ 3 | + 1 | \ 3 / | /
do replacement
___
___ 2*x*\/ 3
v = \/ 3 - ---------
3 then
the integral =
/ | | 1 2* | ------ dv = 2*atan(v) | 2 | 1 + v | /
do backward replacement
/ | / ___\ | 1 ___ | ___ 2*x*\/ 3 | 2* | ------------------------- dx = \/ 3 *atan|- \/ 3 + ---------| | 2 \ 3 / | / ___ \ | |-2*\/ 3 ___| | |--------*x + \/ 3 | + 1 | \ 3 / | /
Solution is:
/ 2 \ / ___\
log\3 + x - 3*x/ ___ | ___ 2*x*\/ 3 |
C + ----------------- + \/ 3 *atan|- \/ 3 + ---------|
2 \ 3 /
The answer (Indefinite)
[src]
/ | / 2 \ / ___ \ | x log\3 + x - 3*x/ ___ |2*\/ 3 *(-3/2 + x)| | ------------ dx = C + ----------------- + \/ 3 *atan|------------------| | 2 2 \ 3 / | x - 3*x + 3 | /
$$\int \frac{x}{\left(x^{2} - 3 x\right) + 3}\, dx = C + \frac{\log{\left(x^{2} - 3 x + 3 \right)}}{2} + \sqrt{3} \operatorname{atan}{\left(\frac{2 \sqrt{3} \left(x - \frac{3}{2}\right)}{3} \right)}$$
The graph
The answer
[src]
___
log(3) pi*\/ 3
- ------ + --------
2 6
$$- \frac{\log{\left(3 \right)}}{2} + \frac{\sqrt{3} \pi}{6}$$
=
=
___
log(3) pi*\/ 3
- ------ + --------
2 6
$$- \frac{\log{\left(3 \right)}}{2} + \frac{\sqrt{3} \pi}{6}$$
-log(3)/2 + pi*sqrt(3)/6
The graph
Use the examples entering the upper and lower limits of integration.