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How to use it?
- Integral of d{x}:
-
Integral of 1/(2+x^2)
- Integral of sin(w*t)^2
- Integral of sin(log(x))^2/x
- Integral of sin^n(x)
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Identical expressions
- (3x+ two)/(x^ two -x+ one)
- (3x plus 2) divide by (x squared minus x plus 1)
- (3x plus two) divide by (x to the power of two minus x plus one)
- (3x+2)/(x2-x+1)
- 3x+2/x2-x+1
- (3x+2)/(x²-x+1)
- (3x+2)/(x to the power of 2-x+1)
- 3x+2/x^2-x+1
- (3x+2) divide by (x^2-x+1)
- (3x+2)/(x^2-x+1)dx
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Similar expressions
- (3x+2)/(x^2-x-1)
- (3x+2)/(x^2+x+1)
- (3x-2)/(x^2-x+1)
Integral of (3x+2)/(x^2-x+1) dx
The solution
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[src]
1 / | | 3*x + 2 | ---------- dx | 2 | x - x + 1 | / 0
$$\int\limits_{0}^{1} \frac{3 x + 2}{x^{2} - x + 1}\, dx$$
Integral((3*x + 2)/(x^2 - x + 1), (x, 0, 1))
Detail solution
We have the integral:
/ | | 3*x + 2 | 1*---------- dx | 2 | x - x + 1 | /
Rewrite the integrand
1*2*x - 1
3*------------ / 7 \
2 |-----|
3*x + 2 1*x - x + 1 \2*3/4/
---------- = -------------- + -------------------------
2 2 2
x - x + 1 / ___ ___\
|-2*\/ 3 \/ 3 |
|--------*x + -----| + 1
\ 3 3 / or
/ | | 3*x + 2 | 1*---------- dx | 2 = | x - x + 1 | /
/
|
| 1
/ 14* | ------------------------- dx
| | 2
| 1*2*x - 1 | / ___ ___\
3* | ------------ dx | |-2*\/ 3 \/ 3 |
| 2 | |--------*x + -----| + 1
| 1*x - x + 1 | \ 3 3 /
| |
/ /
-------------------- + ----------------------------------
2 3 In the integral
/
|
| 1*2*x - 1
3* | ------------ dx
| 2
| 1*x - x + 1
|
/
--------------------
2 do replacement
2 u = x - x
then
the integral =
/
|
| 1
3* | ----- du
| 1 + u
|
/ 3*log(1 + u)
------------- = ------------
2 2 do backward replacement
/
|
| 1*2*x - 1
3* | ------------ dx
| 2
| 1*x - x + 1
| / 2 \
/ 3*log\1 + x - x/
-------------------- = -----------------
2 2 In the integral
/
|
| 1
14* | ------------------------- dx
| 2
| / ___ ___\
| |-2*\/ 3 \/ 3 |
| |--------*x + -----| + 1
| \ 3 3 /
|
/
----------------------------------
3 do replacement
___ ___
\/ 3 2*x*\/ 3
v = ----- - ---------
3 3 then
the integral =
/
|
| 1
14* | ------ dv
| 2
| 1 + v
|
/ 14*atan(v)
--------------- = ----------
3 3 do backward replacement
/
|
| 1
14* | ------------------------- dx
| 2
| / ___ ___\
| |-2*\/ 3 \/ 3 |
| |--------*x + -----| + 1 / ___ ___\
| \ 3 3 / ___ | \/ 3 2*x*\/ 3 |
| 7*\/ 3 *atan|- ----- + ---------|
/ \ 3 3 /
---------------------------------- = ---------------------------------
3 3 Solution is:
/ ___ ___\
___ | \/ 3 2*x*\/ 3 |
/ 2 \ 7*\/ 3 *atan|- ----- + ---------|
3*log\1 + x - x/ \ 3 3 /
C + ----------------- + ---------------------------------
2 3
The answer (Indefinite)
[src]
/ ___ \ / ___ |2*\/ 3 *(-1/2 + x)| | / 2 \ 7*\/ 3 *atan|------------------| | 3*x + 2 3*log\1 + x - x/ \ 3 / | ---------- dx = C + ----------------- + -------------------------------- | 2 2 3 | x - x + 1 | /
$${{3\,\log \left(x^2-x+1\right)}\over{2}}+{{7\,\arctan \left({{2\,x-
1}\over{\sqrt{3}}}\right)}\over{\sqrt{3}}}$$
The graph
The answer
[src]
___
7*pi*\/ 3
----------
9
$${{7\,\pi}\over{3^{{{3}\over{2}}}}}$$
=
=
___
7*pi*\/ 3
----------
9
$$\frac{7 \sqrt{3} \pi}{9}$$
The graph
Use the examples entering the upper and lower limits of integration.