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How to use it?
- Integral of d{x}:
-
Integral of e^(2*x+1)
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Integral of sin(5x)
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Integral of -xe^x
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Integral of x*dx/(x^2-1)
- How do you in partial fractions?:
- (4x-1)/(x^2-4x+8)
-
Identical expressions
- (4x- one)/(x^ two -4x+ eight)
- (4x minus 1) divide by (x squared minus 4x plus 8)
- (4x minus one) divide by (x to the power of two minus 4x plus eight)
- (4x-1)/(x2-4x+8)
- 4x-1/x2-4x+8
- (4x-1)/(x²-4x+8)
- (4x-1)/(x to the power of 2-4x+8)
- 4x-1/x^2-4x+8
- (4x-1) divide by (x^2-4x+8)
- (4x-1)/(x^2-4x+8)dx
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Similar expressions
- (4x-1)/(x^2+4x+8)
- (4x-1)/(x^2-4x-8)
- (4x+1)/(x^2-4x+8)
Integral of (4x-1)/(x^2-4x+8) dx
The solution
You have entered
[src]
1 / | | 4*x - 1 | ------------ dx | 2 | x - 4*x + 8 | / 0
$$\int\limits_{0}^{1} \frac{4 x - 1}{\left(x^{2} - 4 x\right) + 8}\, dx$$
Integral((4*x - 1)/(x^2 - 4*x + 8), (x, 0, 1))
Detail solution
We have the integral:
/ | | 4*x - 1 | ------------ dx | 2 | x - 4*x + 8 | /
Rewrite the integrand
/7\
|-|
4*x - 1 2*x - 4 \4/
------------ = 2*------------ + --------------
2 2 2
x - 4*x + 8 x - 4*x + 8 / x \
|- - + 1| + 1
\ 2 / or
/ | | 4*x - 1 | ------------ dx | 2 = | x - 4*x + 8 | /
/
|
| 1
7* | -------------- dx
| 2
| / x \
| |- - + 1| + 1
/ | \ 2 /
| |
| 2*x - 4 /
2* | ------------ dx + ----------------------
| 2 4
| x - 4*x + 8
|
/ In the integral
/ | | 2*x - 4 2* | ------------ dx | 2 | x - 4*x + 8 | /
do replacement
2 u = x - 4*x
then
the integral =
/ | | 1 2* | ----- du = 2*log(8 + u) | 8 + u | /
do backward replacement
/ | | 2*x - 4 / 2 \ 2* | ------------ dx = 2*log\8 + x - 4*x/ | 2 | x - 4*x + 8 | /
In the integral
/
|
| 1
7* | -------------- dx
| 2
| / x \
| |- - + 1| + 1
| \ 2 /
|
/
----------------------
4 do replacement
x
v = 1 - -
2then
the integral =
/
|
| 1
7* | ------ dv
| 2
| 1 + v
|
/ 7*atan(v)
-------------- = ---------
4 4 do backward replacement
/
|
| 1
7* | -------------- dx
| 2
| / x \
| |- - + 1| + 1
| \ 2 / / x\
| 7*atan|-1 + -|
/ \ 2/
---------------------- = --------------
4 2 Solution is:
/ x\
7*atan|-1 + -|
/ 2 \ \ 2/
C + 2*log\8 + x - 4*x/ + --------------
2
The answer (Indefinite)
[src]
/ / x\ | 7*atan|-1 + -| | 4*x - 1 / 2 \ \ 2/ | ------------ dx = C + 2*log\8 + x - 4*x/ + -------------- | 2 2 | x - 4*x + 8 | /
$$\int \frac{4 x - 1}{\left(x^{2} - 4 x\right) + 8}\, dx = C + 2 \log{\left(x^{2} - 4 x + 8 \right)} + \frac{7 \operatorname{atan}{\left(\frac{x}{2} - 1 \right)}}{2}$$
The graph
The answer
[src]
7*atan(1/2) 7*pi
-2*log(8) + 2*log(5) - ----------- + ----
2 8
$$- 2 \log{\left(8 \right)} - \frac{7 \operatorname{atan}{\left(\frac{1}{2} \right)}}{2} + \frac{7 \pi}{8} + 2 \log{\left(5 \right)}$$
=
=
7*atan(1/2) 7*pi
-2*log(8) + 2*log(5) - ----------- + ----
2 8
$$- 2 \log{\left(8 \right)} - \frac{7 \operatorname{atan}{\left(\frac{1}{2} \right)}}{2} + \frac{7 \pi}{8} + 2 \log{\left(5 \right)}$$
-2*log(8) + 2*log(5) - 7*atan(1/2)/2 + 7*pi/8
Use the examples entering the upper and lower limits of integration.