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- Improper integral
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How to use it?
- Integral of d{x}:
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Integral of 1/(x^2+6x+11)
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Integral of sin10x
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Integral of cos(3x^2+4)
- Integral of sin(4x)*sin(5x)
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Identical expressions
- one /(x^ two +6x+ eleven)
- 1 divide by (x squared plus 6x plus 11)
- one divide by (x to the power of two plus 6x plus eleven)
- 1/(x2+6x+11)
- 1/x2+6x+11
- 1/(x²+6x+11)
- 1/(x to the power of 2+6x+11)
- 1/x^2+6x+11
- 1 divide by (x^2+6x+11)
- 1/(x^2+6x+11)dx
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Similar expressions
- 1/(x^2+6x-11)
- 1/(x^2-6x+11)
Integral of 1/(x^2+6x+11) dx
The solution
You have entered
[src]
oo / | | 1 | 1*------------- dx | 2 | x + 6*x + 11 | / 2
$$\int\limits_{2}^{\infty} 1 \cdot \frac{1}{x^{2} + 6 x + 11}\, dx$$
Integral(1/(x^2 + 6*x + 11), (x, 2, oo))
Detail solution
We have the integral:
/ | | 1 | 1*1*------------- dx | 2 | x + 6*x + 11 | /
Rewrite the integrand
1 1
1*------------- = ------------------------------
2 / 2 \
x + 6*x + 11 |/ ___ ___\ |
||-\/ 2 3*\/ 2 | |
2*||-------*x - -------| + 1|
\\ 2 2 / /or
/ | | 1 | 1*1*------------- dx | 2 = | x + 6*x + 11 | /
/
|
| 1
| -------------------------- dx
| 2
| / ___ ___\
| |-\/ 2 3*\/ 2 |
| |-------*x - -------| + 1
| \ 2 2 /
|
/
--------------------------------
2 In the integral
/
|
| 1
| -------------------------- dx
| 2
| / ___ ___\
| |-\/ 2 3*\/ 2 |
| |-------*x - -------| + 1
| \ 2 2 /
|
/
--------------------------------
2 do replacement
___ ___
3*\/ 2 x*\/ 2
v = - ------- - -------
2 2 then
the integral =
/
|
| 1
| ------ dv
| 2
| 1 + v
|
/ atan(v)
------------ = -------
2 2 do backward replacement
/
|
| 1
| -------------------------- dx
| 2
| / ___ ___\
| |-\/ 2 3*\/ 2 |
| |-------*x - -------| + 1 / ___ ___\
| \ 2 2 / ___ |3*\/ 2 x*\/ 2 |
| \/ 2 *atan|------- + -------|
/ \ 2 2 /
-------------------------------- = -----------------------------
2 2 Solution is:
/ ___ ___\
___ |3*\/ 2 x*\/ 2 |
\/ 2 *atan|------- + -------|
\ 2 2 /
C + -----------------------------
2
The answer (Indefinite)
[src]
/ ___ ___\ / ___ |3*\/ 2 x*\/ 2 | | \/ 2 *atan|------- + -------| | 1 \ 2 2 / | 1*------------- dx = C + ----------------------------- | 2 2 | x + 6*x + 11 | /
$${{\arctan \left({{2\,x+6}\over{2^{{{3}\over{2}}}}}\right)}\over{
\sqrt{2}}}$$
The graph
The answer
[src]
/ ___\
___ |5*\/ 2 |
\/ 2 *atan|-------| ___
\ 2 / pi*\/ 2
- ------------------- + --------
2 4
$$- \frac{\sqrt{2} \operatorname{atan}{\left(\frac{5 \sqrt{2}}{2} \right)}}{2} + \frac{\sqrt{2} \pi}{4}$$
=
=
/ ___\
___ |5*\/ 2 |
\/ 2 *atan|-------| ___
\ 2 / pi*\/ 2
- ------------------- + --------
2 4
$$- \frac{\sqrt{2} \operatorname{atan}{\left(\frac{5 \sqrt{2}}{2} \right)}}{2} + \frac{\sqrt{2} \pi}{4}$$
The graph
Use the examples entering the upper and lower limits of integration.