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- Improper integral
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How to use it?
- Integral of d{x}:
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Integral of (x*ln(x))/(1+x^2)^2
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Integral of x/(x^2+4*x+8)
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Integral of x/(sqrt(2x+1)+1)
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Integral of x*cos6x
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Identical expressions
- x/(x^ two + four *x+ eight)
- x divide by (x squared plus 4 multiply by x plus 8)
- x divide by (x to the power of two plus four multiply by x plus eight)
- x/(x2+4*x+8)
- x/x2+4*x+8
- x/(x²+4*x+8)
- x/(x to the power of 2+4*x+8)
- x/(x^2+4x+8)
- x/(x2+4x+8)
- x/x2+4x+8
- x/x^2+4x+8
- x divide by (x^2+4*x+8)
- x/(x^2+4*x+8)dx
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Similar expressions
- x/(x^2+4*x-8)
- x/(x^2-4*x+8)
Integral of x/(x^2+4*x+8) dx
The solution
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[src]
1 / | | x | ------------ dx | 2 | x + 4*x + 8 | / 0
$$\int\limits_{0}^{1} \frac{x}{x^{2} + 4 x + 8}\, dx$$
Integral(x/(x^2 + 4*x + 8), (x, 0, 1))
Detail solution
We have the integral:
/ | | x | 1*------------ dx | 2 | x + 4*x + 8 | /
Rewrite the integrand
/ 1*2*x + 4 \
|--------------| /-2 \
| 2 | |---|
x \1*x + 4*x + 8/ \ 4 /
------------ = ---------------- + --------------
2 2 2
x + 4*x + 8 / x \
|- - - 1| + 1
\ 2 / or
/ | | x | 1*------------ dx | 2 = | x + 4*x + 8 | /
/
|
/ | 1
| | -------------- dx
| 1*2*x + 4 | 2
| -------------- dx | / x \
| 2 | |- - - 1| + 1
| 1*x + 4*x + 8 | \ 2 /
| |
/ /
-------------------- - --------------------
2 2 In the integral
/
|
| 1*2*x + 4
| -------------- dx
| 2
| 1*x + 4*x + 8
|
/
--------------------
2 do replacement
2 u = x + 4*x
then
the integral =
/
|
| 1
| ----- du
| 8 + u
|
/ log(8 + u)
----------- = ----------
2 2 do backward replacement
/
|
| 1*2*x + 4
| -------------- dx
| 2
| 1*x + 4*x + 8
| / 2 \
/ log\8 + x + 4*x/
-------------------- = -----------------
2 2 In the integral
/
|
| 1
- | -------------- dx
| 2
| / x \
| |- - - 1| + 1
| \ 2 /
|
/
----------------------
2 do replacement
x
v = -1 - -
2then
the integral =
/
|
| 1
- | ------ dv
| 2
| 1 + v
|
/ -atan(v)
-------------- = ---------
2 2 do backward replacement
/
|
| 1
- | -------------- dx
| 2
| / x \
| |- - - 1| + 1
| \ 2 /
|
/ / x\
---------------------- = -atan|1 + -|
2 \ 2/Solution is:
/ 2 \
log\8 + x + 4*x/ / x\
C + ----------------- - atan|1 + -|
2 \ 2/
The answer (Indefinite)
[src]
/ | / 2 \ | x log\8 + x + 4*x/ / x\ | ------------ dx = C + ----------------- - atan|1 + -| | 2 2 \ 2/ | x + 4*x + 8 | /
$$\int \frac{x}{x^{2} + 4 x + 8}\, dx = C + \frac{\log{\left(x^{2} + 4 x + 8 \right)}}{2} - \operatorname{atan}{\left(\frac{x}{2} + 1 \right)}$$
The graph
The answer
[src]
log(13) log(8) pi ------- - atan(3/2) - ------ + -- 2 2 4
$$- \frac{\log{\left(8 \right)}}{2} - \operatorname{atan}{\left(\frac{3}{2} \right)} + \frac{\pi}{4} + \frac{\log{\left(13 \right)}}{2}$$
=
=
log(13) log(8) pi ------- - atan(3/2) - ------ + -- 2 2 4
$$- \frac{\log{\left(8 \right)}}{2} - \operatorname{atan}{\left(\frac{3}{2} \right)} + \frac{\pi}{4} + \frac{\log{\left(13 \right)}}{2}$$
The graph
Use the examples entering the upper and lower limits of integration.