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-
How to use it?
- Integral of d{x}:
-
Integral of 4x²
-
Integral of (x-x^2)
-
Integral of (x*ln(x))/(1+x^2)^2
-
Integral of x/(x^2+4*x+8)
- Canonical form:
- x/(x^2-2x+5)
-
Identical expressions
- x/(x^ two -2x+ five)
- x divide by (x squared minus 2x plus 5)
- x divide by (x to the power of two minus 2x plus five)
- x/(x2-2x+5)
- x/x2-2x+5
- x/(x²-2x+5)
- x/(x to the power of 2-2x+5)
- x/x^2-2x+5
- x divide by (x^2-2x+5)
- x/(x^2-2x+5)dx
-
Similar expressions
- x/(x^2+2x+5)
- x/(x^2-2x-5)
Integral of x/(x^2-2x+5) dx
The solution
You have entered
[src]
1 / | | x | ------------ dx | 2 | x - 2*x + 5 | / 0
$$\int\limits_{0}^{1} \frac{x}{\left(x^{2} - 2 x\right) + 5}\, dx$$
Integral(x/(x^2 - 2*x + 5), (x, 0, 1))
Detail solution
We have the integral:
/ | | x | ------------ dx | 2 | x - 2*x + 5 | /
Rewrite the integrand
/ 2*x - 2 \
|------------|
| 2 |
x \x - 2*x + 5/ 1
------------ = -------------- + ------------------
2 2 / 2 \
x - 2*x + 5 |/ x 1\ |
4*||- - + -| + 1|
\\ 2 2/ /or
/ | | x | ------------ dx | 2 = | x - 2*x + 5 | /
/
|
/ | 1
| | -------------- dx
| 2*x - 2 | 2
| ------------ dx | / x 1\
| 2 | |- - + -| + 1
| x - 2*x + 5 | \ 2 2/
| |
/ /
------------------ + --------------------
2 4 In the integral
/
|
| 2*x - 2
| ------------ dx
| 2
| x - 2*x + 5
|
/
------------------
2 do replacement
2 u = x - 2*x
then
the integral =
/
|
| 1
| ----- du
| 5 + u
|
/ log(5 + u)
----------- = ----------
2 2 do backward replacement
/
|
| 2*x - 2
| ------------ dx
| 2
| x - 2*x + 5
| / 2 \
/ log\5 + x - 2*x/
------------------ = -----------------
2 2 In the integral
/
|
| 1
| -------------- dx
| 2
| / x 1\
| |- - + -| + 1
| \ 2 2/
|
/
--------------------
4 do replacement
1 x
v = - - -
2 2then
the integral =
/
|
| 1
| ------ dv
| 2
| 1 + v
|
/ atan(v)
------------ = -------
4 4 do backward replacement
/
|
| 1
| -------------- dx
| 2
| / x 1\
| |- - + -| + 1
| \ 2 2/ / 1 x\
| atan|- - + -|
/ \ 2 2/
-------------------- = -------------
4 2 Solution is:
/ 1 x\
atan|- - + -| / 2 \
\ 2 2/ log\5 + x - 2*x/
C + ------------- + -----------------
2 2
The answer (Indefinite)
[src]
/ / 1 x\ | atan|- - + -| / 2 \ | x \ 2 2/ log\5 + x - 2*x/ | ------------ dx = C + ------------- + ----------------- | 2 2 2 | x - 2*x + 5 | /
$$\int \frac{x}{\left(x^{2} - 2 x\right) + 5}\, dx = C + \frac{\log{\left(x^{2} - 2 x + 5 \right)}}{2} + \frac{\operatorname{atan}{\left(\frac{x}{2} - \frac{1}{2} \right)}}{2}$$
The graph
The answer
[src]
atan(1/2) log(4) log(5)
--------- + ------ - ------
2 2 2
$$- \frac{\log{\left(5 \right)}}{2} + \frac{\operatorname{atan}{\left(\frac{1}{2} \right)}}{2} + \frac{\log{\left(4 \right)}}{2}$$
=
=
atan(1/2) log(4) log(5)
--------- + ------ - ------
2 2 2
$$- \frac{\log{\left(5 \right)}}{2} + \frac{\operatorname{atan}{\left(\frac{1}{2} \right)}}{2} + \frac{\log{\left(4 \right)}}{2}$$
atan(1/2)/2 + log(4)/2 - log(5)/2
The graph
Use the examples entering the upper and lower limits of integration.