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- Improper integral
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How to use it?
- Integral of d{x}:
-
Integral of 2/x^4
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Integral of (1-x^2)/(1+x^2)
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Integral of 1/(1+4x^2)
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Integral of x*dx/(x^2+1)
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Identical expressions
- x/(x^(two)+x+ one)
- x divide by (x to the power of (2) plus x plus 1)
- x divide by (x to the power of (two) plus x plus one)
- x/(x(2)+x+1)
- x/x2+x+1
- x/x^2+x+1
- x divide by (x^(2)+x+1)
- x/(x^(2)+x+1)dx
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Similar expressions
- x/(x^2+x+1)^(1/3)
- x/(x^(2)+x-1)
- x/(x^(2)-x+1)
Integral of x/(x^(2)+x+1) dx
The solution
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[src]
1 / | | x | ---------- dx | 2 | x + x + 1 | / 0
$$\int\limits_{0}^{1} \frac{x}{x^{2} + x + 1}\, dx$$
Integral(x/(x^2 + x + 1), (x, 0, 1))
Detail solution
We have the integral:
/ | | x | 1*---------- dx | 2 | x + x + 1 | /
Rewrite the integrand
/ 1*2*x + 1 \
|--------------| / -1 \
| 2 | |-----|
x \1*x + 1*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
| 1*2*x + 1 | / ___ ___\
| -------------- dx | |-2*\/ 3 \/ 3 |
| 2 | |--------*x - -----| + 1
| 1*x + 1*x + 1 | \ 3 3 /
| |
/ /
-------------------- - ---------------------------------
2 3 In the integral
/
|
| 1*2*x + 1
| -------------- dx
| 2
| 1*x + 1*x + 1
|
/
--------------------
2 do replacement
2 u = x + x
then
the integral =
/
|
| 1
| ----- du
| 1 + u
|
/ log(1 + u)
----------- = ----------
2 2 do backward replacement
/
|
| 1*2*x + 1
| -------------- dx
| 2
| 1*x + 1*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 log\1 + x + x / \ 3 / | ---------- dx = C + --------------- - ----------------------------- | 2 2 3 | x + x + 1 | /
$${{\log \left(x^2+x+1\right)}\over{2}}-{{\arctan \left({{2\,x+1
}\over{\sqrt{3}}}\right)}\over{\sqrt{3}}}$$
The graph
The answer
[src]
___ log(3) pi*\/ 3 ------ - -------- 2 18
$${{\log 3}\over{2}}-{{\pi}\over{2\,3^{{{3}\over{2}}}}}$$
=
=
___ log(3) pi*\/ 3 ------ - -------- 2 18
$$- \frac{\sqrt{3} \pi}{18} + \frac{\log{\left(3 \right)}}{2}$$
The graph
Use the examples entering the upper and lower limits of integration.