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- Improper integral
- Double Integral
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How to use it?
- Integral of d{x}:
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Integral of x²+4
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Integral of x^3e^x
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Integral of 2x^3+1
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Integral of 2*x^2+2/x^4
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Identical expressions
- one /(6x^ two -3x+ two)
- 1 divide by (6x squared minus 3x plus 2)
- one divide by (6x to the power of two minus 3x plus two)
- 1/(6x2-3x+2)
- 1/6x2-3x+2
- 1/(6x²-3x+2)
- 1/(6x to the power of 2-3x+2)
- 1/6x^2-3x+2
- 1 divide by (6x^2-3x+2)
- 1/(6x^2-3x+2)dx
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Similar expressions
- 1/(6x^2-3x-2)
- 1/(6x^2+3x+2)
Integral of 1/(6x^2-3x+2) dx
The solution
You have entered
[src]
1 / | | 1 | -------------- dx | 2 | 6*x - 3*x + 2 | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(6 x^{2} - 3 x\right) + 2}\, dx$$
Integral(1/(6*x^2 - 3*x + 2), (x, 0, 1))
Detail solution
We have the integral:
/ | | 1 | -------------- dx | 2 | 6*x - 3*x + 2 | /
Rewrite the integrand
1 1
-------------- = ----------------------------------
2 / 2 \
6*x - 3*x + 2 |/ ____ ____\ |
||-4*\/ 39 \/ 39 | |
13/8*||---------*x + ------| + 1|
\\ 13 13 / /or
/ | | 1 | -------------- dx | 2 = | 6*x - 3*x + 2 | /
/
|
| 1
8* | --------------------------- dx
| 2
| / ____ ____\
| |-4*\/ 39 \/ 39 |
| |---------*x + ------| + 1
| \ 13 13 /
|
/
-----------------------------------
13 In the integral
/
|
| 1
8* | --------------------------- dx
| 2
| / ____ ____\
| |-4*\/ 39 \/ 39 |
| |---------*x + ------| + 1
| \ 13 13 /
|
/
-----------------------------------
13 do replacement
____ ____
\/ 39 4*x*\/ 39
v = ------ - ----------
13 13 then
the integral =
/
|
| 1
8* | ------ dv
| 2
| 1 + v
|
/ 8*atan(v)
-------------- = ---------
13 13 do backward replacement
/
|
| 1
8* | --------------------------- dx
| 2
| / ____ ____\
| |-4*\/ 39 \/ 39 |
| |---------*x + ------| + 1 / ____ ____\
| \ 13 13 / ____ | \/ 39 4*x*\/ 39 |
| 2*\/ 39 *atan|- ------ + ----------|
/ \ 13 13 /
----------------------------------- = ------------------------------------
13 39 Solution is:
/ ____ ____\
____ | \/ 39 4*x*\/ 39 |
2*\/ 39 *atan|- ------ + ----------|
\ 13 13 /
C + ------------------------------------
39
The answer (Indefinite)
[src]
/ ____ \ / ____ |4*\/ 39 *(-1/4 + x)| | 2*\/ 39 *atan|-------------------| | 1 \ 13 / | -------------- dx = C + ---------------------------------- | 2 39 | 6*x - 3*x + 2 | /
$$\int \frac{1}{\left(6 x^{2} - 3 x\right) + 2}\, dx = C + \frac{2 \sqrt{39} \operatorname{atan}{\left(\frac{4 \sqrt{39} \left(x - \frac{1}{4}\right)}{13} \right)}}{39}$$
The graph
The answer
[src]
/ ____\ / ____\
____ |\/ 39 | ____ |3*\/ 39 |
2*\/ 39 *atan|------| 2*\/ 39 *atan|--------|
\ 13 / \ 13 /
--------------------- + -----------------------
39 39
$$\frac{2 \sqrt{39} \operatorname{atan}{\left(\frac{\sqrt{39}}{13} \right)}}{39} + \frac{2 \sqrt{39} \operatorname{atan}{\left(\frac{3 \sqrt{39}}{13} \right)}}{39}$$
=
=
/ ____\ / ____\
____ |\/ 39 | ____ |3*\/ 39 |
2*\/ 39 *atan|------| 2*\/ 39 *atan|--------|
\ 13 / \ 13 /
--------------------- + -----------------------
39 39
$$\frac{2 \sqrt{39} \operatorname{atan}{\left(\frac{\sqrt{39}}{13} \right)}}{39} + \frac{2 \sqrt{39} \operatorname{atan}{\left(\frac{3 \sqrt{39}}{13} \right)}}{39}$$
2*sqrt(39)*atan(sqrt(39)/13)/39 + 2*sqrt(39)*atan(3*sqrt(39)/13)/39
Use the examples entering the upper and lower limits of integration.