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How to use it?
- Integral of d{x}:
-
Integral of -1/y
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Integral of 1/4x
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Integral of x/(x-2)
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Integral of x/(x^2+2x+10)
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Identical expressions
- x/(x^ two +2x+ ten)
- x divide by (x squared plus 2x plus 10)
- x divide by (x to the power of two plus 2x plus ten)
- x/(x2+2x+10)
- x/x2+2x+10
- x/(x²+2x+10)
- x/(x to the power of 2+2x+10)
- x/x^2+2x+10
- x divide by (x^2+2x+10)
- x/(x^2+2x+10)dx
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Similar expressions
- x/(x^2-2x+10)
- x/(x^2+2x-10)
Integral of x/(x^2+2x+10) dx
The solution
You have entered
[src]
1 / | | x | ------------- dx | 2 | x + 2*x + 10 | / 0
$$\int\limits_{0}^{1} \frac{x}{\left(x^{2} + 2 x\right) + 10}\, dx$$
Integral(x/(x^2 + 2*x + 10), (x, 0, 1))
Detail solution
We have the integral:
/ | | x | ------------- dx | 2 | x + 2*x + 10 | /
Rewrite the integrand
/ 2*x + 2 \
|-------------| /-1 \
| 2 | |---|
x \x + 2*x + 10/ \ 9 /
------------- = --------------- + --------------
2 2 2
x + 2*x + 10 / x 1\
|- - - -| + 1
\ 3 3/ or
/ | | x | ------------- dx | 2 = | x + 2*x + 10 | /
/
|
/ | 1
| | -------------- dx
| 2*x + 2 | 2
| ------------- dx | / x 1\
| 2 | |- - - -| + 1
| x + 2*x + 10 | \ 3 3/
| |
/ /
------------------- - --------------------
2 9 In the integral
/
|
| 2*x + 2
| ------------- dx
| 2
| x + 2*x + 10
|
/
-------------------
2 do replacement
2 u = x + 2*x
then
the integral =
/
|
| 1
| ------ du
| 10 + u
|
/ log(10 + u)
------------ = -----------
2 2 do backward replacement
/
|
| 2*x + 2
| ------------- dx
| 2
| x + 2*x + 10
| / 2 \
/ log\10 + x + 2*x/
------------------- = ------------------
2 2 In the integral
/
|
| 1
- | -------------- dx
| 2
| / x 1\
| |- - - -| + 1
| \ 3 3/
|
/
----------------------
9 do replacement
1 x
v = - - - -
3 3then
the integral =
/
|
| 1
- | ------ dv
| 2
| 1 + v
|
/ -atan(v)
-------------- = ---------
9 9 do backward replacement
/
|
| 1
- | -------------- dx
| 2
| / x 1\
| |- - - -| + 1
| \ 3 3/ /1 x\
| -atan|- + -|
/ \3 3/
---------------------- = -------------
9 3 Solution is:
/1 x\
/ 2 \ atan|- + -|
log\10 + x + 2*x/ \3 3/
C + ------------------ - -----------
2 3
The answer (Indefinite)
[src]
/ /1 x\ | / 2 \ atan|- + -| | x log\10 + x + 2*x/ \3 3/ | ------------- dx = C + ------------------ - ----------- | 2 2 3 | x + 2*x + 10 | /
$$\int \frac{x}{\left(x^{2} + 2 x\right) + 10}\, dx = C + \frac{\log{\left(x^{2} + 2 x + 10 \right)}}{2} - \frac{\operatorname{atan}{\left(\frac{x}{3} + \frac{1}{3} \right)}}{3}$$
The graph
The answer
[src]
log(13) log(10) atan(2/3) atan(1/3) ------- - ------- - --------- + --------- 2 2 3 3
$$- \frac{\log{\left(10 \right)}}{2} - \frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{3} + \frac{\operatorname{atan}{\left(\frac{1}{3} \right)}}{3} + \frac{\log{\left(13 \right)}}{2}$$
=
=
log(13) log(10) atan(2/3) atan(1/3) ------- - ------- - --------- + --------- 2 2 3 3
$$- \frac{\log{\left(10 \right)}}{2} - \frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{3} + \frac{\operatorname{atan}{\left(\frac{1}{3} \right)}}{3} + \frac{\log{\left(13 \right)}}{2}$$
log(13)/2 - log(10)/2 - atan(2/3)/3 + atan(1/3)/3
The graph
Use the examples entering the upper and lower limits of integration.