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- Improper integral
- Double Integral
- Triple Integral
- Curvilinear integral
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- Differential equations Step by Step
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How to use it?
- Integral of d{x}:
-
Integral of x^4/(1+x^2)
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Integral of x^5/(x^2+1)
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Integral of (tan(x))^2/(cos(x))^2
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Integral of sqrt(x^2+1)*dx/x
- How do you in partial fractions?:
- (x-2)/(x^2-x+1)
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Identical expressions
- (x- two)/(x^ two -x+ one)
- (x minus 2) divide by (x squared minus x plus 1)
- (x minus two) divide by (x to the power of two minus x plus one)
- (x-2)/(x2-x+1)
- x-2/x2-x+1
- (x-2)/(x²-x+1)
- (x-2)/(x to the power of 2-x+1)
- x-2/x^2-x+1
- (x-2) divide by (x^2-x+1)
- (x-2)/(x^2-x+1)dx
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Similar expressions
- (x+2)/(x^2-x+1)
- (x-2)/(x^2-x-1)
- (x-2)/(x^2+x+1)
Integral of (x-2)/(x^2-x+1) dx
The solution
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[src]
1 / | | x - 2 | ---------- dx | 2 | x - x + 1 | / 0
$$\int\limits_{0}^{1} \frac{x - 2}{\left(x^{2} - x\right) + 1}\, dx$$
Integral((x - 2)/(x^2 - x + 1), (x, 0, 1))
Detail solution
We have the integral:
/ | | x - 2 | ---------- dx | 2 | x - x + 1 | /
Rewrite the integrand
/ 2*x - 1 \
|----------| / -3 \
| 2 | |-----|
x - 2 \x - x + 1/ \2*3/4/
---------- = ------------ + -------------------------
2 2 2
x - x + 1 / ___ ___\
|-2*\/ 3 \/ 3 |
|--------*x + -----| + 1
\ 3 3 / or
/ | | x - 2 | ---------- dx | 2 = | x - x + 1 | /
/
|
| 2*x - 1
| ---------- dx
| 2
| x - x + 1 /
| |
/ | 1
---------------- - 2* | ------------------------- dx
2 | 2
| / ___ ___\
| |-2*\/ 3 \/ 3 |
| |--------*x + -----| + 1
| \ 3 3 /
|
/ In the integral
/
|
| 2*x - 1
| ---------- dx
| 2
| x - x + 1
|
/
----------------
2 do replacement
2 u = x - x
then
the integral =
/
|
| 1
| ----- du
| 1 + u
|
/ log(1 + u)
----------- = ----------
2 2 do backward replacement
/
|
| 2*x - 1
| ---------- dx
| 2
| x - x + 1
| / 2 \
/ log\1 + x - x/
---------------- = ---------------
2 2 In the integral
/
|
| 1
-2* | ------------------------- dx
| 2
| / ___ ___\
| |-2*\/ 3 \/ 3 |
| |--------*x + -----| + 1
| \ 3 3 /
|
/ do replacement
___ ___
\/ 3 2*x*\/ 3
v = ----- - ---------
3 3 then
the integral =
/
|
| 1
-2* | ------ dv = -2*atan(v)
| 2
| 1 + v
|
/ do backward replacement
/
| / ___ ___\
| 1 ___ | \/ 3 2*x*\/ 3 |
-2* | ------------------------- dx = -\/ 3 *atan|- ----- + ---------|
| 2 \ 3 3 /
| / ___ ___\
| |-2*\/ 3 \/ 3 |
| |--------*x + -----| + 1
| \ 3 3 /
|
/ Solution is:
/ 2 \ / ___ ___\
log\1 + x - x/ ___ | \/ 3 2*x*\/ 3 |
C + --------------- - \/ 3 *atan|- ----- + ---------|
2 \ 3 3 /
The answer (Indefinite)
[src]
/ | / 2 \ / ___ \ | x - 2 log\1 + x - x/ ___ |2*\/ 3 *(-1/2 + x)| | ---------- dx = C + --------------- - \/ 3 *atan|------------------| | 2 2 \ 3 / | x - x + 1 | /
$$\int \frac{x - 2}{\left(x^{2} - x\right) + 1}\, dx = C + \frac{\log{\left(x^{2} - x + 1 \right)}}{2} - \sqrt{3} \operatorname{atan}{\left(\frac{2 \sqrt{3} \left(x - \frac{1}{2}\right)}{3} \right)}$$
The graph
The answer
[src]
___
-pi*\/ 3
----------
3
$$- \frac{\sqrt{3} \pi}{3}$$
=
=
___
-pi*\/ 3
----------
3
$$- \frac{\sqrt{3} \pi}{3}$$
-pi*sqrt(3)/3
The graph
Use the examples entering the upper and lower limits of integration.