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How to use it?
- Integral of d{x}:
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Integral of 1/(2+x^2)
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Integral of x*sin2x
- Integral of x/(x^3-3x+2)
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Integral of -x+4
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Identical expressions
- (x)/((x)^ two - two *x+ four)
- (x) divide by ((x) squared minus 2 multiply by x plus 4)
- (x) divide by ((x) to the power of two minus two multiply by x plus four)
- (x)/((x)2-2*x+4)
- x/x2-2*x+4
- (x)/((x)²-2*x+4)
- (x)/((x) to the power of 2-2*x+4)
- (x)/((x)^2-2x+4)
- (x)/((x)2-2x+4)
- x/x2-2x+4
- x/x^2-2x+4
- (x) divide by ((x)^2-2*x+4)
- (x)/((x)^2-2*x+4)dx
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Similar expressions
- (x)/((x)^2-2*x-4)
- (x)/((x)^2+2*x+4)
Integral of (x)/((x)^2-2*x+4) dx
The solution
You have entered
[src]
1 / | | x | ------------ dx | 2 | x - 2*x + 4 | / 0
$$\int\limits_{0}^{1} \frac{x}{\left(x^{2} - 2 x\right) + 4}\, dx$$
Integral(x/(x^2 - 2*x + 4), (x, 0, 1))
Detail solution
We have the integral:
/ | | x | ------------ dx | 2 | x - 2*x + 4 | /
Rewrite the integrand
/ 2*x - 2 \
|------------|
| 2 |
x \x - 2*x + 4/ 1
------------ = -------------- + ----------------------------
2 2 / 2 \
x - 2*x + 4 |/ ___ ___\ |
||-\/ 3 \/ 3 | |
3*||-------*x + -----| + 1|
\\ 3 3 / /or
/ | | x | ------------ dx | 2 = | x - 2*x + 4 | /
/
|
| 1
/ | ------------------------ dx
| | 2
| 2*x - 2 | / ___ ___\
| ------------ dx | |-\/ 3 \/ 3 |
| 2 | |-------*x + -----| + 1
| x - 2*x + 4 | \ 3 3 /
| |
/ /
------------------ + ------------------------------
2 3 In the integral
/
|
| 2*x - 2
| ------------ dx
| 2
| x - 2*x + 4
|
/
------------------
2 do replacement
2 u = x - 2*x
then
the integral =
/
|
| 1
| ----- du
| 4 + u
|
/ log(4 + u)
----------- = ----------
2 2 do backward replacement
/
|
| 2*x - 2
| ------------ dx
| 2
| x - 2*x + 4
| / 2 \
/ log\4 + x - 2*x/
------------------ = -----------------
2 2 In the integral
/
|
| 1
| ------------------------ dx
| 2
| / ___ ___\
| |-\/ 3 \/ 3 |
| |-------*x + -----| + 1
| \ 3 3 /
|
/
------------------------------
3 do replacement
___ ___
\/ 3 x*\/ 3
v = ----- - -------
3 3 then
the integral =
/
|
| 1
| ------ dv
| 2
| 1 + v
|
/ atan(v)
------------ = -------
3 3 do backward replacement
/
|
| 1
| ------------------------ dx
| 2
| / ___ ___\
| |-\/ 3 \/ 3 |
| |-------*x + -----| + 1 / ___ ___\
| \ 3 3 / ___ | \/ 3 x*\/ 3 |
| \/ 3 *atan|- ----- + -------|
/ \ 3 3 /
------------------------------ = -----------------------------
3 3 Solution is:
/ ___ ___\
___ | \/ 3 x*\/ 3 |
/ 2 \ \/ 3 *atan|- ----- + -------|
log\4 + x - 2*x/ \ 3 3 /
C + ----------------- + -----------------------------
2 3
The answer (Indefinite)
[src]
/ ___ \ / ___ |\/ 3 *(-1 + x)| | / 2 \ \/ 3 *atan|--------------| | x log\4 + x - 2*x/ \ 3 / | ------------ dx = C + ----------------- + -------------------------- | 2 2 3 | x - 2*x + 4 | /
$$\int \frac{x}{\left(x^{2} - 2 x\right) + 4}\, dx = C + \frac{\log{\left(x^{2} - 2 x + 4 \right)}}{2} + \frac{\sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \left(x - 1\right)}{3} \right)}}{3}$$
The graph
The answer
[src]
___ log(3) log(4) pi*\/ 3 ------ - ------ + -------- 2 2 18
$$- \frac{\log{\left(4 \right)}}{2} + \frac{\sqrt{3} \pi}{18} + \frac{\log{\left(3 \right)}}{2}$$
=
=
___ log(3) log(4) pi*\/ 3 ------ - ------ + -------- 2 2 18
$$- \frac{\log{\left(4 \right)}}{2} + \frac{\sqrt{3} \pi}{18} + \frac{\log{\left(3 \right)}}{2}$$
log(3)/2 - log(4)/2 + pi*sqrt(3)/18
Use the examples entering the upper and lower limits of integration.