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How to use it?
- Integral of d{x}:
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Integral of x×e^x
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Integral of 1/(x²+1)
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Integral of 1/√(1-4x^2)
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Integral of x^2e^x
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Identical expressions
- (3x+ five)/(x^ two + nine)
- (3x plus 5) divide by (x squared plus 9)
- (3x plus five) divide by (x to the power of two plus nine)
- (3x+5)/(x2+9)
- 3x+5/x2+9
- (3x+5)/(x²+9)
- (3x+5)/(x to the power of 2+9)
- 3x+5/x^2+9
- (3x+5) divide by (x^2+9)
- (3x+5)/(x^2+9)dx
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Similar expressions
- (3x+5)/(x^2-9)
- (3x-5)/(x^2+9)
Integral of (3x+5)/(x^2+9) dx
The solution
You have entered
[src]
oo / | | 3*x + 5 | ------- dx | 2 | x + 9 | / -oo
$$\int\limits_{-\infty}^{\infty} \frac{3 x + 5}{x^{2} + 9}\, dx$$
Integral((3*x + 5)/(x^2 + 9), (x, -oo, oo))
Detail solution
We have the integral:
/ | | 3*x + 5 | ------- dx | 2 | x + 9 | /
Rewrite the integrand
2*x
3*------------ /5\
2 |-|
3*x + 5 x + 0*x + 9 \9/
------- = -------------- + ----------
2 2 2
x + 9 /-x \
|---| + 1
\ 3 / or
/ | | 3*x + 5 | ------- dx | 2 = | x + 9 | /
/
|
/ | 1
| 5* | ---------- dx
| 2*x | 2
3* | ------------ dx | /-x \
| 2 | |---| + 1
| x + 0*x + 9 | \ 3 /
| |
/ /
-------------------- + ------------------
2 9 In the integral
/
|
| 2*x
3* | ------------ dx
| 2
| x + 0*x + 9
|
/
--------------------
2 do replacement
2 u = x
then
the integral =
/
|
| 1
3* | ----- du
| 9 + u
|
/ 3*log(9 + u)
------------- = ------------
2 2 do backward replacement
/
|
| 2*x
3* | ------------ dx
| 2
| x + 0*x + 9
| / 2\
/ 3*log\9 + x /
-------------------- = -------------
2 2 In the integral
/
|
| 1
5* | ---------- dx
| 2
| /-x \
| |---| + 1
| \ 3 /
|
/
------------------
9 do replacement
-x
v = ---
3 then
the integral =
/
|
| 1
5* | ------ dv
| 2
| 1 + v
|
/ 5*atan(v)
-------------- = ---------
9 9 do backward replacement
/
|
| 1
5* | ---------- dx
| 2
| /-x \
| |---| + 1
| \ 3 / /x\
| 5*atan|-|
/ \3/
------------------ = ---------
9 3 Solution is:
/x\
/ 2\ 5*atan|-|
3*log\9 + x / \3/
C + ------------- + ---------
2 3
The answer (Indefinite)
[src]
/ /x\ | / 2\ 5*atan|-| | 3*x + 5 3*log\9 + x / \3/ | ------- dx = C + ------------- + --------- | 2 2 3 | x + 9 | /
$$\int \frac{3 x + 5}{x^{2} + 9}\, dx = C + \frac{3 \log{\left(x^{2} + 9 \right)}}{2} + \frac{5 \operatorname{atan}{\left(\frac{x}{3} \right)}}{3}$$
Use the examples entering the upper and lower limits of integration.