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