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