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- Improper integral
- Double Integral
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How to use it?
- Integral of d{x}:
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Integral of x³
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Integral of xarctanx
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Integral of ln(2x)
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Integral of 2x^3
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Identical expressions
- dx/(x^ two - five *x+ eight)
- dx divide by (x squared minus 5 multiply by x plus 8)
- dx divide by (x to the power of two minus five multiply by x plus eight)
- dx/(x2-5*x+8)
- dx/x2-5*x+8
- dx/(x²-5*x+8)
- dx/(x to the power of 2-5*x+8)
- dx/(x^2-5x+8)
- dx/(x2-5x+8)
- dx/x2-5x+8
- dx/x^2-5x+8
- dx divide by (x^2-5*x+8)
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Similar expressions
- dx/(x^2+5*x+8)
- dx/(x^2-5*x-8)
Integral of dx/(x^2-5*x+8) dx
The solution
You have entered
[src]
1 / | | 1 | ------------ dx | 2 | x - 5*x + 8 | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(x^{2} - 5 x\right) + 8}\, dx$$
Integral(1/(x^2 - 5*x + 8), (x, 0, 1))
Detail solution
We have the integral:
/ | | 1 | ------------ dx | 2 | x - 5*x + 8 | /
Rewrite the integrand
1 1
------------ = ---------------------------------
2 / 2 \
x - 5*x + 8 |/ ___ ___\ |
||-2*\/ 7 5*\/ 7 | |
7/4*||--------*x + -------| + 1|
\\ 7 7 / /or
/ | | 1 | ------------ dx | 2 = | x - 5*x + 8 | /
/
|
| 1
4* | --------------------------- dx
| 2
| / ___ ___\
| |-2*\/ 7 5*\/ 7 |
| |--------*x + -------| + 1
| \ 7 7 /
|
/
-----------------------------------
7 In the integral
/
|
| 1
4* | --------------------------- dx
| 2
| / ___ ___\
| |-2*\/ 7 5*\/ 7 |
| |--------*x + -------| + 1
| \ 7 7 /
|
/
-----------------------------------
7 do replacement
___ ___
5*\/ 7 2*x*\/ 7
v = ------- - ---------
7 7 then
the integral =
/
|
| 1
4* | ------ dv
| 2
| 1 + v
|
/ 4*atan(v)
-------------- = ---------
7 7 do backward replacement
/
|
| 1
4* | --------------------------- dx
| 2
| / ___ ___\
| |-2*\/ 7 5*\/ 7 |
| |--------*x + -------| + 1 / ___ ___\
| \ 7 7 / ___ | 5*\/ 7 2*x*\/ 7 |
| 2*\/ 7 *atan|- ------- + ---------|
/ \ 7 7 /
----------------------------------- = -----------------------------------
7 7 Solution is:
/ ___ ___\
___ | 5*\/ 7 2*x*\/ 7 |
2*\/ 7 *atan|- ------- + ---------|
\ 7 7 /
C + -----------------------------------
7
The answer (Indefinite)
[src]
/ ___ \ / ___ |2*\/ 7 *(-5/2 + x)| | 2*\/ 7 *atan|------------------| | 1 \ 7 / | ------------ dx = C + -------------------------------- | 2 7 | x - 5*x + 8 | /
$$\int \frac{1}{\left(x^{2} - 5 x\right) + 8}\, dx = C + \frac{2 \sqrt{7} \operatorname{atan}{\left(\frac{2 \sqrt{7} \left(x - \frac{5}{2}\right)}{7} \right)}}{7}$$
The graph
The answer
[src]
/ ___\ / ___\
___ |3*\/ 7 | ___ |5*\/ 7 |
2*\/ 7 *atan|-------| 2*\/ 7 *atan|-------|
\ 7 / \ 7 /
- --------------------- + ---------------------
7 7
$$- \frac{2 \sqrt{7} \operatorname{atan}{\left(\frac{3 \sqrt{7}}{7} \right)}}{7} + \frac{2 \sqrt{7} \operatorname{atan}{\left(\frac{5 \sqrt{7}}{7} \right)}}{7}$$
=
=
/ ___\ / ___\
___ |3*\/ 7 | ___ |5*\/ 7 |
2*\/ 7 *atan|-------| 2*\/ 7 *atan|-------|
\ 7 / \ 7 /
- --------------------- + ---------------------
7 7
$$- \frac{2 \sqrt{7} \operatorname{atan}{\left(\frac{3 \sqrt{7}}{7} \right)}}{7} + \frac{2 \sqrt{7} \operatorname{atan}{\left(\frac{5 \sqrt{7}}{7} \right)}}{7}$$
-2*sqrt(7)*atan(3*sqrt(7)/7)/7 + 2*sqrt(7)*atan(5*sqrt(7)/7)/7
Use the examples entering the upper and lower limits of integration.