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How to use it?
- Integral of d{x}:
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Integral of (x+3)e^x
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Integral of (x(x^2-7))/(x^2+2)dx
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Integral of x/(x^2-6x+13)
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Integral of x*sqrt(x)/x^(3/4)
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Identical expressions
- x/(x^ two -6x+ thirteen)
- x divide by (x squared minus 6x plus 13)
- x divide by (x to the power of two minus 6x plus thirteen)
- x/(x2-6x+13)
- x/x2-6x+13
- x/(x²-6x+13)
- x/(x to the power of 2-6x+13)
- x/x^2-6x+13
- x divide by (x^2-6x+13)
- x/(x^2-6x+13)dx
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Similar expressions
- x/(x^2-6x-13)
- x/(x^2+6x+13)
Integral of x/(x^2-6x+13) dx
The solution
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[src]
1 / | | x | ------------- dx | 2 | x - 6*x + 13 | / 0
$$\int\limits_{0}^{1} \frac{x}{\left(x^{2} - 6 x\right) + 13}\, dx$$
Integral(x/(x^2 - 6*x + 13), (x, 0, 1))
Detail solution
We have the integral:
/ | | x | ------------- dx | 2 | x - 6*x + 13 | /
Rewrite the integrand
/ 2*x - 6 \
|-------------| /3\
| 2 | |-|
x \x - 6*x + 13/ \4/
------------- = --------------- + --------------
2 2 2
x - 6*x + 13 / x 3\
|- - + -| + 1
\ 2 2/ or
/ | | x | ------------- dx | 2 = | x - 6*x + 13 | /
/
|
/ | 1
| 3* | -------------- dx
| 2*x - 6 | 2
| ------------- dx | / x 3\
| 2 | |- - + -| + 1
| x - 6*x + 13 | \ 2 2/
| |
/ /
------------------- + ----------------------
2 4 In the integral
/
|
| 2*x - 6
| ------------- dx
| 2
| x - 6*x + 13
|
/
-------------------
2 do replacement
2 u = x - 6*x
then
the integral =
/
|
| 1
| ------ du
| 13 + u
|
/ log(13 + u)
------------ = -----------
2 2 do backward replacement
/
|
| 2*x - 6
| ------------- dx
| 2
| x - 6*x + 13
| / 2 \
/ log\13 + x - 6*x/
------------------- = ------------------
2 2 In the integral
/
|
| 1
3* | -------------- dx
| 2
| / x 3\
| |- - + -| + 1
| \ 2 2/
|
/
----------------------
4 do replacement
3 x
v = - - -
2 2then
the integral =
/
|
| 1
3* | ------ dv
| 2
| 1 + v
|
/ 3*atan(v)
-------------- = ---------
4 4 do backward replacement
/
|
| 1
3* | -------------- dx
| 2
| / x 3\
| |- - + -| + 1
| \ 2 2/ / 3 x\
| 3*atan|- - + -|
/ \ 2 2/
---------------------- = ---------------
4 2 Solution is:
/ 3 x\
/ 2 \ 3*atan|- - + -|
log\13 + x - 6*x/ \ 2 2/
C + ------------------ + ---------------
2 2
The answer (Indefinite)
[src]
/ / 3 x\ | / 2 \ 3*atan|- - + -| | x log\13 + x - 6*x/ \ 2 2/ | ------------- dx = C + ------------------ + --------------- | 2 2 2 | x - 6*x + 13 | /
$$\int \frac{x}{\left(x^{2} - 6 x\right) + 13}\, dx = C + \frac{\log{\left(x^{2} - 6 x + 13 \right)}}{2} + \frac{3 \operatorname{atan}{\left(\frac{x}{2} - \frac{3}{2} \right)}}{2}$$
The graph
The answer
[src]
log(8) 3*pi log(13) 3*atan(3/2) ------ - ---- - ------- + ----------- 2 8 2 2
$$- \frac{\log{\left(13 \right)}}{2} - \frac{3 \pi}{8} + \frac{\log{\left(8 \right)}}{2} + \frac{3 \operatorname{atan}{\left(\frac{3}{2} \right)}}{2}$$
=
=
log(8) 3*pi log(13) 3*atan(3/2) ------ - ---- - ------- + ----------- 2 8 2 2
$$- \frac{\log{\left(13 \right)}}{2} - \frac{3 \pi}{8} + \frac{\log{\left(8 \right)}}{2} + \frac{3 \operatorname{atan}{\left(\frac{3}{2} \right)}}{2}$$
log(8)/2 - 3*pi/8 - log(13)/2 + 3*atan(3/2)/2
The graph
Use the examples entering the upper and lower limits of integration.