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How to use it?
- Integral of d{x}:
-
Integral of 1/(x(1-x))
-
Integral of sin^3(x)/cos^2(x)
- Integral of ln(x+(1+x^2)^(1/2))
- Integral of e^x^2(ln2^(x/2))
-
Identical expressions
- (x- one)/(x^ two -x+ one)
- (x minus 1) divide by (x squared minus x plus 1)
- (x minus one) divide by (x to the power of two minus x plus one)
- (x-1)/(x2-x+1)
- x-1/x2-x+1
- (x-1)/(x²-x+1)
- (x-1)/(x to the power of 2-x+1)
- x-1/x^2-x+1
- (x-1) divide by (x^2-x+1)
- (x-1)/(x^2-x+1)dx
-
Similar expressions
- (x-1)/(x^2-x-1)
- (x+1)/(x^2-x+1)
- (x-1)/(x^2+x+1)
Integral of (x-1)/(x^2-x+1) dx
The solution
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[src]
1 / | | x - 1 | ---------- dx | 2 | x - x + 1 | / 0
$$\int\limits_{0}^{1} \frac{x - 1}{\left(x^{2} - x\right) + 1}\, dx$$
Integral((x - 1)/(x^2 - x + 1), (x, 0, 1))
Detail solution
We have the integral:
/ | | x - 1 | ---------- dx | 2 | x - x + 1 | /
Rewrite the integrand
/ 2*x - 1 \
|----------| / -1 \
| 2 | |-----|
x - 1 \x - x + 1/ \2*3/4/
---------- = ------------ + -------------------------
2 2 2
x - x + 1 / ___ ___\
|-2*\/ 3 \/ 3 |
|--------*x + -----| + 1
\ 3 3 / or
/ | | x - 1 | ---------- dx | 2 = | x - x + 1 | /
/
|
| 1
/ 2* | ------------------------- dx
| | 2
| 2*x - 1 | / ___ ___\
| ---------- dx | |-2*\/ 3 \/ 3 |
| 2 | |--------*x + -----| + 1
| x - x + 1 | \ 3 3 /
| |
/ /
---------------- - ---------------------------------
2 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 /
|
/
----------------------------------
3 do replacement
___ ___
\/ 3 2*x*\/ 3
v = ----- - ---------
3 3 then
the integral =
/
|
| 1
-2* | ------ dv
| 2
| 1 + v
|
/ -2*atan(v)
--------------- = ----------
3 3 do backward replacement
/
|
| 1
-2* | ------------------------- dx
| 2
| / ___ ___\
| |-2*\/ 3 \/ 3 |
| |--------*x + -----| + 1 / ___ ___\
| \ 3 3 / ___ | \/ 3 2*x*\/ 3 |
| -\/ 3 *atan|- ----- + ---------|
/ \ 3 3 /
---------------------------------- = ---------------------------------
3 3 Solution is:
/ ___ ___\
___ | \/ 3 2*x*\/ 3 |
/ 2 \ \/ 3 *atan|- ----- + ---------|
log\1 + x - x/ \ 3 3 /
C + --------------- - -------------------------------
2 3
The answer (Indefinite)
[src]
/ ___ \ / ___ |2*\/ 3 *(-1/2 + x)| | / 2 \ \/ 3 *atan|------------------| | x - 1 log\1 + x - x/ \ 3 / | ---------- dx = C + --------------- - ------------------------------ | 2 2 3 | x - x + 1 | /
$$\int \frac{x - 1}{\left(x^{2} - x\right) + 1}\, dx = C + \frac{\log{\left(x^{2} - x + 1 \right)}}{2} - \frac{\sqrt{3} \operatorname{atan}{\left(\frac{2 \sqrt{3} \left(x - \frac{1}{2}\right)}{3} \right)}}{3}$$
The graph
The answer
[src]
___
-pi*\/ 3
----------
9
$$- \frac{\sqrt{3} \pi}{9}$$
=
=
___
-pi*\/ 3
----------
9
$$- \frac{\sqrt{3} \pi}{9}$$
-pi*sqrt(3)/9
The graph
Use the examples entering the upper and lower limits of integration.