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- Improper integral
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How to use it?
- Integral of d{x}:
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Integral of (1+2x^2)/(x^2(1+x^2))
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Integral of 1/100
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Integral of x^5*dx
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Integral of xsin(x+2)
- Derivative of:
- (x+2)/(x^2+1)
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Identical expressions
- (x+ two)/(x^ two + one)
- (x plus 2) divide by (x squared plus 1)
- (x plus 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]
1 / | | x + 2 | ------ dx | 2 | x + 1 | / 0
$$\int\limits_{0}^{1} \frac{x + 2}{x^{2} + 1}\, dx$$
Integral((x + 2)/(x^2 + 1), (x, 0, 1))
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
The answer
[src]
pi log(2) -- + ------ 2 2
$$\frac{\log{\left(2 \right)}}{2} + \frac{\pi}{2}$$
=
=
pi log(2) -- + ------ 2 2
$$\frac{\log{\left(2 \right)}}{2} + \frac{\pi}{2}$$
pi/2 + log(2)/2
Use the examples entering the upper and lower limits of integration.