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