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