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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/(x^2+2*x+5)
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Integral of 1/sqrt(x^2+1)
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Integral of 4*x*exp(x^2)
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Integral of (sec(x))^3
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Identical expressions
- x/(x^ two -5x+ seven)
- x divide by (x squared minus 5x plus 7)
- x divide by (x to the power of two minus 5x plus seven)
- x/(x2-5x+7)
- x/x2-5x+7
- x/(x²-5x+7)
- x/(x to the power of 2-5x+7)
- x/x^2-5x+7
- x divide by (x^2-5x+7)
- x/(x^2-5x+7)dx
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Similar expressions
- x/(x^2-5x-7)
- x/(x^2+5x+7)
Integral of x/(x^2-5x+7) dx
The solution
You have entered
[src]
1 / | | x | ------------ dx | 2 | x - 5*x + 7 | / 0
$$\int\limits_{0}^{1} \frac{x}{\left(x^{2} - 5 x\right) + 7}\, dx$$
Integral(x/(x^2 - 5*x + 7), (x, 0, 1))
Detail solution
We have the integral:
/ | | x | ------------ dx | 2 | x - 5*x + 7 | /
Rewrite the integrand
/ 2*x - 5 \
|------------| / 5 \
| 2 | |-----|
x \x - 5*x + 7/ \2*3/4/
------------ = -------------- + ---------------------------
2 2 2
x - 5*x + 7 / ___ ___\
|-2*\/ 3 5*\/ 3 |
|--------*x + -------| + 1
\ 3 3 / or
/ | | x | ------------ dx | 2 = | x - 5*x + 7 | /
/
|
| 1
/ 10* | --------------------------- dx
| | 2
| 2*x - 5 | / ___ ___\
| ------------ dx | |-2*\/ 3 5*\/ 3 |
| 2 | |--------*x + -------| + 1
| x - 5*x + 7 | \ 3 3 /
| |
/ /
------------------ + ------------------------------------
2 3 In the integral
/
|
| 2*x - 5
| ------------ dx
| 2
| x - 5*x + 7
|
/
------------------
2 do replacement
2 u = x - 5*x
then
the integral =
/
|
| 1
| ----- du
| 7 + u
|
/ log(7 + u)
----------- = ----------
2 2 do backward replacement
/
|
| 2*x - 5
| ------------ dx
| 2
| x - 5*x + 7
| / 2 \
/ log\7 + x - 5*x/
------------------ = -----------------
2 2 In the integral
/
|
| 1
10* | --------------------------- dx
| 2
| / ___ ___\
| |-2*\/ 3 5*\/ 3 |
| |--------*x + -------| + 1
| \ 3 3 /
|
/
------------------------------------
3 do replacement
___ ___
5*\/ 3 2*x*\/ 3
v = ------- - ---------
3 3 then
the integral =
/
|
| 1
10* | ------ dv
| 2
| 1 + v
|
/ 10*atan(v)
--------------- = ----------
3 3 do backward replacement
/
|
| 1
10* | --------------------------- dx
| 2
| / ___ ___\
| |-2*\/ 3 5*\/ 3 |
| |--------*x + -------| + 1 / ___ ___\
| \ 3 3 / ___ | 5*\/ 3 2*x*\/ 3 |
| 5*\/ 3 *atan|- ------- + ---------|
/ \ 3 3 /
------------------------------------ = -----------------------------------
3 3 Solution is:
/ ___ ___\
___ | 5*\/ 3 2*x*\/ 3 |
/ 2 \ 5*\/ 3 *atan|- ------- + ---------|
log\7 + x - 5*x/ \ 3 3 /
C + ----------------- + -----------------------------------
2 3
The answer (Indefinite)
[src]
/ ___ \ / ___ |2*\/ 3 *(-5/2 + x)| | / 2 \ 5*\/ 3 *atan|------------------| | x log\7 + x - 5*x/ \ 3 / | ------------ dx = C + ----------------- + -------------------------------- | 2 2 3 | x - 5*x + 7 | /
$$\int \frac{x}{\left(x^{2} - 5 x\right) + 7}\, dx = C + \frac{\log{\left(x^{2} - 5 x + 7 \right)}}{2} + \frac{5 \sqrt{3} \operatorname{atan}{\left(\frac{2 \sqrt{3} \left(x - \frac{5}{2}\right)}{3} \right)}}{3}$$
The graph
The answer
[src]
/ ___\
___ |5*\/ 3 |
___ 5*\/ 3 *atan|-------|
log(3) log(7) 5*pi*\/ 3 \ 3 /
------ - ------ - ---------- + ---------------------
2 2 9 3
$$- \frac{5 \sqrt{3} \pi}{9} - \frac{\log{\left(7 \right)}}{2} + \frac{\log{\left(3 \right)}}{2} + \frac{5 \sqrt{3} \operatorname{atan}{\left(\frac{5 \sqrt{3}}{3} \right)}}{3}$$
=
=
/ ___\
___ |5*\/ 3 |
___ 5*\/ 3 *atan|-------|
log(3) log(7) 5*pi*\/ 3 \ 3 /
------ - ------ - ---------- + ---------------------
2 2 9 3
$$- \frac{5 \sqrt{3} \pi}{9} - \frac{\log{\left(7 \right)}}{2} + \frac{\log{\left(3 \right)}}{2} + \frac{5 \sqrt{3} \operatorname{atan}{\left(\frac{5 \sqrt{3}}{3} \right)}}{3}$$
log(3)/2 - log(7)/2 - 5*pi*sqrt(3)/9 + 5*sqrt(3)*atan(5*sqrt(3)/3)/3
The graph
Use the examples entering the upper and lower limits of integration.