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How to use it?
- Integral of d{x}:
-
Integral of x*e^(4*x)
- Integral of x/sqrt(x^2-4x+1)
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Integral of x/(x^2-1)^3
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Integral of x*cos2xdx
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Identical expressions
- (x- two)/(x^ two +x+ one)
- (x minus 2) divide by (x squared plus x plus 1)
- (x minus two) divide by (x to the power of two plus 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)
- (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 \
|----------| / -5 \
| 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 | /
/
|
| 1
/ 10* | ------------------------- 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
-10* | ------------------------- dx
| 2
| / ___ ___\
| |-2*\/ 3 \/ 3 |
| |--------*x - -----| + 1
| \ 3 3 /
|
/
-----------------------------------
3 do replacement
___ ___
\/ 3 2*x*\/ 3
v = - ----- - ---------
3 3 then
the integral =
/
|
| 1
-10* | ------ dv
| 2
| 1 + v
|
/ -10*atan(v)
---------------- = -----------
3 3 do backward replacement
/
|
| 1
-10* | ------------------------- dx
| 2
| / ___ ___\
| |-2*\/ 3 \/ 3 |
| |--------*x - -----| + 1 / ___ ___\
| \ 3 3 / ___ |\/ 3 2*x*\/ 3 |
| -5*\/ 3 *atan|----- + ---------|
/ \ 3 3 /
----------------------------------- = --------------------------------
3 3 Solution is:
/ ___ ___\
___ |\/ 3 2*x*\/ 3 |
/ 2\ 5*\/ 3 *atan|----- + ---------|
log\1 + x + x / \ 3 3 /
C + --------------- - -------------------------------
2 3
The answer (Indefinite)
[src]
/ ___ \ / ___ |2*\/ 3 *(1/2 + x)| | / 2\ 5*\/ 3 *atan|-----------------| | x - 2 log\1 + x + x / \ 3 / | ---------- dx = C + --------------- - ------------------------------- | 2 2 3 | x + x + 1 | /
$$\int \frac{x - 2}{\left(x^{2} + x\right) + 1}\, dx = C + \frac{\log{\left(x^{2} + x + 1 \right)}}{2} - \frac{5 \sqrt{3} \operatorname{atan}{\left(\frac{2 \sqrt{3} \left(x + \frac{1}{2}\right)}{3} \right)}}{3}$$
The graph
The answer
[src]
___ log(3) 5*pi*\/ 3 ------ - ---------- 2 18
$$- \frac{5 \sqrt{3} \pi}{18} + \frac{\log{\left(3 \right)}}{2}$$
=
=
___ log(3) 5*pi*\/ 3 ------ - ---------- 2 18
$$- \frac{5 \sqrt{3} \pi}{18} + \frac{\log{\left(3 \right)}}{2}$$
log(3)/2 - 5*pi*sqrt(3)/18
The graph
Use the examples entering the upper and lower limits of integration.