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- Improper integral
- Double Integral
- Triple Integral
- Curvilinear integral
- Derivative Step by Step
- Differential equations Step by Step
- Limits Step by Step
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How to use it?
- Integral of d{x}:
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Integral of e^(-x)/sqrt(x)
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Integral of 8/x^2
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Integral of (1+x^2)^0.5
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Integral of 2*t
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Identical expressions
- one /(x^ two -2x+ nine)
- 1 divide by (x squared minus 2x plus 9)
- one divide by (x to the power of two minus 2x plus nine)
- 1/(x2-2x+9)
- 1/x2-2x+9
- 1/(x²-2x+9)
- 1/(x to the power of 2-2x+9)
- 1/x^2-2x+9
- 1 divide by (x^2-2x+9)
- 1/(x^2-2x+9)dx
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Similar expressions
- 1/(x^2-2x-9)
- 1/(x^2+2x+9)
Integral of 1/(x^2-2x+9) dx
The solution
You have entered
[src]
oo / | | 1 | 1*------------ dx | 2 | x - 2*x + 9 | / 0
$$\int\limits_{0}^{\infty} 1 \cdot \frac{1}{x^{2} - 2 x + 9}\, dx$$
Detail solution
We have the integral:
/ | | 1 | 1*1*------------ dx | 2 | x - 2*x + 9 | /
Rewrite the integrand
1 1
1*------------ = ----------------------------
2 / 2 \
x - 2*x + 9 |/ ___ ___\ |
||-\/ 2 \/ 2 | |
8*||-------*x + -----| + 1|
\\ 4 4 / /or
/ | | 1 | 1*1*------------ dx | 2 = | x - 2*x + 9 | /
/
|
| 1
| ------------------------ dx
| 2
| / ___ ___\
| |-\/ 2 \/ 2 |
| |-------*x + -----| + 1
| \ 4 4 /
|
/
------------------------------
8 In the integral
/
|
| 1
| ------------------------ dx
| 2
| / ___ ___\
| |-\/ 2 \/ 2 |
| |-------*x + -----| + 1
| \ 4 4 /
|
/
------------------------------
8 do replacement
___ ___
\/ 2 x*\/ 2
v = ----- - -------
4 4 then
the integral =
/
|
| 1
| ------ dv
| 2
| 1 + v
|
/ atan(v)
------------ = -------
8 8 do backward replacement
/
|
| 1
| ------------------------ dx
| 2
| / ___ ___\
| |-\/ 2 \/ 2 |
| |-------*x + -----| + 1 / ___ ___\
| \ 4 4 / ___ | \/ 2 x*\/ 2 |
| \/ 2 *atan|- ----- + -------|
/ \ 4 4 /
------------------------------ = -----------------------------
8 4 Solution is:
/ ___ ___\
___ | \/ 2 x*\/ 2 |
\/ 2 *atan|- ----- + -------|
\ 4 4 /
C + -----------------------------
4
The answer (Indefinite)
[src]
/ ___ \ / ___ |\/ 2 *(-1 + x)| | \/ 2 *atan|--------------| | 1 \ 4 / | 1*------------ dx = C + -------------------------- | 2 4 | x - 2*x + 9 | /
$${{\arctan \left({{2\,x-2}\over{2^{{{5}\over{2}}}}}\right)}\over{2^{
{{3}\over{2}}}}}$$
The graph
The answer
[src]
/ ___\
___ |\/ 2 |
\/ 2 *atan|-----| ___
\ 4 / pi*\/ 2
----------------- + --------
4 8
$$\frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2}}{4} \right)}}{4} + \frac{\sqrt{2} \pi}{8}$$
=
=
/ ___\
___ |\/ 2 |
\/ 2 *atan|-----| ___
\ 4 / pi*\/ 2
----------------- + --------
4 8
$$\frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2}}{4} \right)}}{4} + \frac{\sqrt{2} \pi}{8}$$
The graph
Use the examples entering the upper and lower limits of integration.