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- Improper integral
- Double Integral
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- Curvilinear integral
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How to use it?
- Integral of d{x}:
-
Integral of 1/(x²+1)
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Integral of f(x)=4x
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Integral of x/sin^2(x)
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Integral of x/cosx^2
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Identical expressions
- x/(x²+4x+ five)
- x divide by (x² plus 4x plus 5)
- x divide by (x² plus 4x plus five)
- x/x²+4x+5
- x divide by (x²+4x+5)
- x/(x²+4x+5)dx
-
Similar expressions
- x/(x²+4x-5)
- x/(x²-4x+5)
Integral of x/(x²+4x+5) dx
The solution
You have entered
[src]
1 / | | x | ------------ dx | 2 | x + 4*x + 5 | / 0
$$\int\limits_{0}^{1} \frac{x}{x^{2} + 4 x + 5}\, dx$$
Integral(x/(x^2 + 4*x + 5), (x, 0, 1))
Detail solution
We have the integral:
/ | | x | 1*------------ dx | 2 | x + 4*x + 5 | /
Rewrite the integrand
/ 1*2*x + 4 \
|--------------| /-2 \
| 2 | |---|
x \1*x + 4*x + 5/ \ 1 /
------------ = ---------------- + -------------
2 2 2
x + 4*x + 5 (-x - 2) + 1or
/ | | x | 1*------------ dx | 2 = | x + 4*x + 5 | /
/
|
| 1*2*x + 4
| -------------- dx
| 2
| 1*x + 4*x + 5 /
| |
/ | 1
-------------------- - 2* | ------------- dx
2 | 2
| (-x - 2) + 1
|
/ In the integral
/
|
| 1*2*x + 4
| -------------- dx
| 2
| 1*x + 4*x + 5
|
/
--------------------
2 do replacement
2 u = x + 4*x
then
the integral =
/
|
| 1
| ----- du
| 5 + u
|
/ log(5 + u)
----------- = ----------
2 2 do backward replacement
/
|
| 1*2*x + 4
| -------------- dx
| 2
| 1*x + 4*x + 5
| / 2 \
/ log\5 + x + 4*x/
-------------------- = -----------------
2 2 In the integral
/
|
| 1
-2* | ------------- dx
| 2
| (-x - 2) + 1
|
/ do replacement
v = -2 - x
then
the integral =
/
|
| 1
-2* | ------ dv = -2*atan(v)
| 2
| 1 + v
|
/ do backward replacement
/
|
| 1
-2* | ------------- dx = -2*atan(2 + x)
| 2
| (-x - 2) + 1
|
/ Solution is:
/ 2 \
log\5 + x + 4*x/
C + ----------------- - 2*atan(2 + x)
2
The answer (Indefinite)
[src]
/ | / 2 \ | x log\5 + x + 4*x/ | ------------ dx = C + ----------------- - 2*atan(2 + x) | 2 2 | x + 4*x + 5 | /
$${{\log \left(x^2+4\,x+5\right)}\over{2}}-2\,\arctan \left({{2\,x+4
}\over{2}}\right)$$
The graph
The answer
[src]
log(10) log(5) ------- - 2*atan(3) + 2*atan(2) - ------ 2 2
$${{\log 10}\over{2}}-{{\log 5}\over{2}}-2\,\arctan 3+2\,\arctan 2$$
=
=
log(10) log(5) ------- - 2*atan(3) + 2*atan(2) - ------ 2 2
$$- 2 \operatorname{atan}{\left(3 \right)} - \frac{\log{\left(5 \right)}}{2} + \frac{\log{\left(10 \right)}}{2} + 2 \operatorname{atan}{\left(2 \right)}$$
The graph
Use the examples entering the upper and lower limits of integration.