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