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