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How to use it?
- Integral of d{x}:
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Integral of sin(x²)
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Integral of x/(x^2+x)
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Integral of xcos(1-x)
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Integral of (x+3)/(x²+2x+5)
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Identical expressions
- (x+ three)/(x²+2x+ five)
- (x plus 3) divide by (x² plus 2x plus 5)
- (x plus three) divide by (x² plus 2x plus five)
- x+3/x²+2x+5
- (x+3) divide by (x²+2x+5)
- (x+3)/(x²+2x+5)dx
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Similar expressions
- (x+3)/(x²-2x+5)
- (x+3)/(x²+2x-5)
- (x-3)/(x²+2x+5)
Integral of (x+3)/(x²+2x+5) dx
The solution
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[src]
1 / | | x + 3 | ------------ dx | 2 | x + 2*x + 5 | / 0
$$\int\limits_{0}^{1} \frac{x + 3}{\left(x^{2} + 2 x\right) + 5}\, dx$$
Integral((x + 3)/(x^2 + 2*x + 5), (x, 0, 1))
Detail solution
We have the integral:
/ | | x + 3 | ------------ dx | 2 | x + 2*x + 5 | /
Rewrite the integrand
/ 2*x + 2 \
|------------| /2\
| 2 | |-|
x + 3 \x + 2*x + 5/ \4/
------------ = -------------- + --------------
2 2 2
x + 2*x + 5 / x 1\
|- - - -| + 1
\ 2 2/ or
/ | | x + 3 | ------------ dx | 2 = | x + 2*x + 5 | /
/
|
| 1 /
| -------------- dx |
| 2 | 2*x + 2
| / x 1\ | ------------ dx
| |- - - -| + 1 | 2
| \ 2 2/ | x + 2*x + 5
| |
/ /
-------------------- + ------------------
2 2 In the integral
/
|
| 2*x + 2
| ------------ dx
| 2
| x + 2*x + 5
|
/
------------------
2 do replacement
2 u = x + 2*x
then
the integral =
/
|
| 1
| ----- du
| 5 + u
|
/ log(5 + u)
----------- = ----------
2 2 do backward replacement
/
|
| 2*x + 2
| ------------ dx
| 2
| x + 2*x + 5
| / 2 \
/ log\5 + x + 2*x/
------------------ = -----------------
2 2 In the integral
/
|
| 1
| -------------- dx
| 2
| / x 1\
| |- - - -| + 1
| \ 2 2/
|
/
--------------------
2 do replacement
1 x
v = - - - -
2 2then
the integral =
/
|
| 1
| ------ dv
| 2
| 1 + v
|
/ atan(v)
------------ = -------
2 2 do backward replacement
/
|
| 1
| -------------- dx
| 2
| / x 1\
| |- - - -| + 1
| \ 2 2/
|
/ /1 x\
-------------------- = atan|- + -|
2 \2 2/Solution is:
/ 2 \
log\5 + x + 2*x/ /1 x\
C + ----------------- + atan|- + -|
2 \2 2/
The answer (Indefinite)
[src]
/ | / 2 \ | x + 3 log\5 + x + 2*x/ /1 x\ | ------------ dx = C + ----------------- + atan|- + -| | 2 2 \2 2/ | x + 2*x + 5 | /
$$\int \frac{x + 3}{\left(x^{2} + 2 x\right) + 5}\, dx = C + \frac{\log{\left(x^{2} + 2 x + 5 \right)}}{2} + \operatorname{atan}{\left(\frac{x}{2} + \frac{1}{2} \right)}$$
The graph
The answer
[src]
log(8) log(5) pi ------ - atan(1/2) - ------ + -- 2 2 4
$$- \frac{\log{\left(5 \right)}}{2} - \operatorname{atan}{\left(\frac{1}{2} \right)} + \frac{\pi}{4} + \frac{\log{\left(8 \right)}}{2}$$
=
=
log(8) log(5) pi ------ - atan(1/2) - ------ + -- 2 2 4
$$- \frac{\log{\left(5 \right)}}{2} - \operatorname{atan}{\left(\frac{1}{2} \right)} + \frac{\pi}{4} + \frac{\log{\left(8 \right)}}{2}$$
log(8)/2 - atan(1/2) - log(5)/2 + pi/4
The graph
Use the examples entering the upper and lower limits of integration.