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- Improper integral
- Double Integral
- Triple Integral
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- Differential equations Step by Step
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How to use it?
- Integral of d{x}:
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Integral of -1/y
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Integral of e^(-x/2)
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Integral of (x^2+1)e^-x
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Integral of 4^(2*x)
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Identical expressions
- one /(x^ two +6x+ twelve)
- 1 divide by (x squared plus 6x plus 12)
- one divide by (x to the power of two plus 6x plus twelve)
- 1/(x2+6x+12)
- 1/x2+6x+12
- 1/(x²+6x+12)
- 1/(x to the power of 2+6x+12)
- 1/x^2+6x+12
- 1 divide by (x^2+6x+12)
- 1/(x^2+6x+12)dx
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Similar expressions
- 1/(x^2-6x+12)
- 1/(x^2+6x-12)
Integral of 1/(x^2+6x+12) dx
The solution
You have entered
[src]
1 / | | 1 | ------------- dx | 2 | x + 6*x + 12 | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(x^{2} + 6 x\right) + 12}\, dx$$
Integral(1/(x^2 + 6*x + 12), (x, 0, 1))
Detail solution
We have the integral:
/ | | 1 | ------------- dx | 2 | x + 6*x + 12 | /
Rewrite the integrand
1 1
------------- = ----------------------------
2 / 2 \
x + 6*x + 12 |/ ___ \ |
||-\/ 3 ___| |
3*||-------*x - \/ 3 | + 1|
\\ 3 / /or
/ | | 1 | ------------- dx | 2 = | x + 6*x + 12 | /
/
|
| 1
| ------------------------ dx
| 2
| / ___ \
| |-\/ 3 ___|
| |-------*x - \/ 3 | + 1
| \ 3 /
|
/
------------------------------
3 In the integral
/
|
| 1
| ------------------------ dx
| 2
| / ___ \
| |-\/ 3 ___|
| |-------*x - \/ 3 | + 1
| \ 3 /
|
/
------------------------------
3 do replacement
___
___ x*\/ 3
v = - \/ 3 - -------
3 then
the integral =
/
|
| 1
| ------ dv
| 2
| 1 + v
|
/ atan(v)
------------ = -------
3 3 do backward replacement
/
|
| 1
| ------------------------ dx
| 2
| / ___ \
| |-\/ 3 ___|
| |-------*x - \/ 3 | + 1 / ___\
| \ 3 / ___ | ___ x*\/ 3 |
| \/ 3 *atan|\/ 3 + -------|
/ \ 3 /
------------------------------ = ---------------------------
3 3 Solution is:
/ ___\
___ | ___ x*\/ 3 |
\/ 3 *atan|\/ 3 + -------|
\ 3 /
C + ---------------------------
3
The answer (Indefinite)
[src]
/ ___ \ / ___ |\/ 3 *(3 + x)| | \/ 3 *atan|-------------| | 1 \ 3 / | ------------- dx = C + ------------------------- | 2 3 | x + 6*x + 12 | /
$$\int \frac{1}{\left(x^{2} + 6 x\right) + 12}\, dx = C + \frac{\sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \left(x + 3\right)}{3} \right)}}{3}$$
The graph
The answer
[src]
/ ___\
___ |4*\/ 3 |
___ \/ 3 *atan|-------|
pi*\/ 3 \ 3 /
- -------- + -------------------
9 3
$$- \frac{\sqrt{3} \pi}{9} + \frac{\sqrt{3} \operatorname{atan}{\left(\frac{4 \sqrt{3}}{3} \right)}}{3}$$
=
=
/ ___\
___ |4*\/ 3 |
___ \/ 3 *atan|-------|
pi*\/ 3 \ 3 /
- -------- + -------------------
9 3
$$- \frac{\sqrt{3} \pi}{9} + \frac{\sqrt{3} \operatorname{atan}{\left(\frac{4 \sqrt{3}}{3} \right)}}{3}$$
-pi*sqrt(3)/9 + sqrt(3)*atan(4*sqrt(3)/3)/3
The graph
Use the examples entering the upper and lower limits of integration.