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How to use it?
- Integral of d{x}:
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Integral of sin(5x)
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Integral of x/(x^3+8)
- Integral of 1/(x+lnx)
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Integral of 1/(2+x^1/2)
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Identical expressions
- one /(three *x^ two -x+ one)
- 1 divide by (3 multiply by x squared minus x plus 1)
- one divide by (three multiply by x to the power of two minus x plus one)
- 1/(3*x2-x+1)
- 1/3*x2-x+1
- 1/(3*x²-x+1)
- 1/(3*x to the power of 2-x+1)
- 1/(3x^2-x+1)
- 1/(3x2-x+1)
- 1/3x2-x+1
- 1/3x^2-x+1
- 1 divide by (3*x^2-x+1)
- 1/(3*x^2-x+1)dx
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Similar expressions
- 1/(3*x^2+x+1)
- 1/(3*x^2-x-1)
Integral of 1/(3*x^2-x+1) dx
The solution
You have entered
[src]
2 / | | 1 | ------------ dx | 2 | 3*x - x + 1 | / 0
$$\int\limits_{0}^{2} \frac{1}{\left(3 x^{2} - x\right) + 1}\, dx$$
Integral(1/(3*x^2 - x + 1), (x, 0, 2))
Detail solution
We have the integral:
/ | | 1 | ------------ dx | 2 | 3*x - x + 1 | /
Rewrite the integrand
1 1
------------ = --------------------------------
2 / 2 \
3*x - x + 1 |/ ____ ____\ |
11 ||-6*\/ 11 \/ 11 | |
--*||---------*x + ------| + 1|
12 \\ 11 11 / /or
/ | | 1 | ------------ dx | 2 = | 3*x - x + 1 | /
/
|
| 1
12* | --------------------------- dx
| 2
| / ____ ____\
| |-6*\/ 11 \/ 11 |
| |---------*x + ------| + 1
| \ 11 11 /
|
/
------------------------------------
11 In the integral
/
|
| 1
12* | --------------------------- dx
| 2
| / ____ ____\
| |-6*\/ 11 \/ 11 |
| |---------*x + ------| + 1
| \ 11 11 /
|
/
------------------------------------
11 do replacement
____ ____
\/ 11 6*x*\/ 11
v = ------ - ----------
11 11 then
the integral =
/
|
| 1
12* | ------ dv
| 2
| 1 + v
|
/ 12*atan(v)
--------------- = ----------
11 11 do backward replacement
/
|
| 1
12* | --------------------------- dx
| 2
| / ____ ____\
| |-6*\/ 11 \/ 11 |
| |---------*x + ------| + 1 / ____ ____\
| \ 11 11 / ____ | \/ 11 6*x*\/ 11 |
| 2*\/ 11 *atan|- ------ + ----------|
/ \ 11 11 /
------------------------------------ = ------------------------------------
11 11 Solution is:
/ ____ ____\
____ | \/ 11 6*x*\/ 11 |
2*\/ 11 *atan|- ------ + ----------|
\ 11 11 /
C + ------------------------------------
11
The answer (Indefinite)
[src]
/ ____ \ / ____ |6*\/ 11 *(-1/6 + x)| | 2*\/ 11 *atan|-------------------| | 1 \ 11 / | ------------ dx = C + ---------------------------------- | 2 11 | 3*x - x + 1 | /
$$\int \frac{1}{\left(3 x^{2} - x\right) + 1}\, dx = C + \frac{2 \sqrt{11} \operatorname{atan}{\left(\frac{6 \sqrt{11} \left(x - \frac{1}{6}\right)}{11} \right)}}{11}$$
The graph
The answer
[src]
/ ____\
____ |\/ 11 |
____ / ____\ 2*\/ 11 *atan|------|
2*\/ 11 *atan\\/ 11 / \ 11 /
--------------------- + ---------------------
11 11
$$\frac{2 \sqrt{11} \operatorname{atan}{\left(\frac{\sqrt{11}}{11} \right)}}{11} + \frac{2 \sqrt{11} \operatorname{atan}{\left(\sqrt{11} \right)}}{11}$$
=
=
/ ____\
____ |\/ 11 |
____ / ____\ 2*\/ 11 *atan|------|
2*\/ 11 *atan\\/ 11 / \ 11 /
--------------------- + ---------------------
11 11
$$\frac{2 \sqrt{11} \operatorname{atan}{\left(\frac{\sqrt{11}}{11} \right)}}{11} + \frac{2 \sqrt{11} \operatorname{atan}{\left(\sqrt{11} \right)}}{11}$$
2*sqrt(11)*atan(sqrt(11))/11 + 2*sqrt(11)*atan(sqrt(11)/11)/11
Use the examples entering the upper and lower limits of integration.