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