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How to use it?
- Integral of d{x}:
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Integral of tan^3(x)dx
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Integral of dx/e^(2*x)
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Integral of xsqrt(1-x^2)
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Integral of xe^(-x)*dx
- 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 plus 2) divide by (x squared plus x plus 1)
- (x plus 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)
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]
___ 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.