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