Mister Exam
Other calculators
- Improper integral
- Double Integral
- Triple Integral
- Curvilinear integral
- Derivative Step by Step
- Differential equations Step by Step
- Limits Step by Step
-
How to use it?
- Integral of d{x}:
-
Integral of x^2/x
-
Integral of cos^2(4x)
-
Integral of (2x-3)^4
-
Integral of -2cosx
-
Identical expressions
- x/(x^ two +2x+ five)
- x divide by (x squared plus 2x plus 5)
- x divide by (x to the power of two plus 2x plus five)
- x/(x2+2x+5)
- x/x2+2x+5
- x/(x²+2x+5)
- x/(x to the power of 2+2x+5)
- x/x^2+2x+5
- x divide by (x^2+2x+5)
- x/(x^2+2x+5)dx
-
Similar expressions
- x/(x^2-2x+5)
- x/(x^2+2x-5)
Integral of x/(x^2+2x+5) dx
The solution
You have entered
[src]
1 / | | x | ------------ dx | 2 | x + 2*x + 5 | / 0
$$\int\limits_{0}^{1} \frac{x}{\left(x^{2} + 2 x\right) + 5}\, dx$$
Integral(x/(x^2 + 2*x + 5), (x, 0, 1))
Detail solution
We have the integral:
/ | | x | ------------ dx | 2 | x + 2*x + 5 | /
Rewrite the integrand
/ 2*x + 2 \
|------------| /-1 \
| 2 | |---|
x \x + 2*x + 5/ \ 4 /
------------ = -------------- + --------------
2 2 2
x + 2*x + 5 / x 1\
|- - - -| + 1
\ 2 2/ or
/ | | x | ------------ dx | 2 = | x + 2*x + 5 | /
/
|
/ | 1
| | -------------- dx
| 2*x + 2 | 2
| ------------ dx | / x 1\
| 2 | |- - - -| + 1
| x + 2*x + 5 | \ 2 2/
| |
/ /
------------------ - --------------------
2 4 In the integral
/
|
| 2*x + 2
| ------------ dx
| 2
| x + 2*x + 5
|
/
------------------
2 do replacement
2 u = x + 2*x
then
the integral =
/
|
| 1
| ----- du
| 5 + u
|
/ log(5 + u)
----------- = ----------
2 2 do backward replacement
/
|
| 2*x + 2
| ------------ dx
| 2
| x + 2*x + 5
| / 2 \
/ log\5 + x + 2*x/
------------------ = -----------------
2 2 In the integral
/
|
| 1
- | -------------- dx
| 2
| / x 1\
| |- - - -| + 1
| \ 2 2/
|
/
----------------------
4 do replacement
1 x
v = - - - -
2 2then
the integral =
/
|
| 1
- | ------ dv
| 2
| 1 + v
|
/ -atan(v)
-------------- = ---------
4 4 do backward replacement
/
|
| 1
- | -------------- dx
| 2
| / x 1\
| |- - - -| + 1
| \ 2 2/ /1 x\
| -atan|- + -|
/ \2 2/
---------------------- = -------------
4 2 Solution is:
/1 x\
/ 2 \ atan|- + -|
log\5 + x + 2*x/ \2 2/
C + ----------------- - -----------
2 2
The answer (Indefinite)
[src]
/ /1 x\ | / 2 \ atan|- + -| | x log\5 + x + 2*x/ \2 2/ | ------------ dx = C + ----------------- - ----------- | 2 2 2 | x + 2*x + 5 | /
$$\int \frac{x}{\left(x^{2} + 2 x\right) + 5}\, dx = C + \frac{\log{\left(x^{2} + 2 x + 5 \right)}}{2} - \frac{\operatorname{atan}{\left(\frac{x}{2} + \frac{1}{2} \right)}}{2}$$
The graph
The answer
[src]
atan(1/2) log(8) log(5) pi
--------- + ------ - ------ - --
2 2 2 8
$$- \frac{\log{\left(5 \right)}}{2} - \frac{\pi}{8} + \frac{\operatorname{atan}{\left(\frac{1}{2} \right)}}{2} + \frac{\log{\left(8 \right)}}{2}$$
=
=
atan(1/2) log(8) log(5) pi
--------- + ------ - ------ - --
2 2 2 8
$$- \frac{\log{\left(5 \right)}}{2} - \frac{\pi}{8} + \frac{\operatorname{atan}{\left(\frac{1}{2} \right)}}{2} + \frac{\log{\left(8 \right)}}{2}$$
atan(1/2)/2 + log(8)/2 - log(5)/2 - pi/8
The graph
Use the examples entering the upper and lower limits of integration.