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