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