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- Improper integral
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How to use it?
- Integral of d{x}:
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Integral of √(x+4)
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Integral of sin³xcosx
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Integral of sin^3xcos^-5x
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Integral of dx/(x^2+8)
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Identical expressions
- dx/(x^ two + eight)
- dx divide by (x squared plus 8)
- dx divide by (x to the power of two plus eight)
- dx/(x2+8)
- dx/x2+8
- dx/(x²+8)
- dx/(x to the power of 2+8)
- dx/x^2+8
- dx divide by (x^2+8)
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Similar expressions
- dx/(x^2-8)
Integral of dx/(x^2+8) dx
The solution
You have entered
[src]
1 / | | 1 | ------ dx | 2 | x + 8 | / 0
$$\int\limits_{0}^{1} \frac{1}{x^{2} + 8}\, dx$$
Integral(1/(x^2 + 8), (x, 0, 1))
Detail solution
We have the integral:
/ | | 1 | ------ dx | 2 | x + 8 | /
Rewrite the integrand
1 1
------ = --------------------
2 / 2 \
x + 8 |/ ___ \ |
||-\/ 2 | |
8*||-------*x| + 1|
\\ 4 / /or
/ | | 1 | ------ dx | 2 = | x + 8 | /
/
|
| 1
| ---------------- dx
| 2
| / ___ \
| |-\/ 2 |
| |-------*x| + 1
| \ 4 /
|
/
----------------------
8 In the integral
/
|
| 1
| ---------------- dx
| 2
| / ___ \
| |-\/ 2 |
| |-------*x| + 1
| \ 4 /
|
/
----------------------
8 do replacement
___
-x*\/ 2
v = ---------
4 then
the integral =
/
|
| 1
| ------ dv
| 2
| 1 + v
|
/ atan(v)
------------ = -------
8 8 do backward replacement
/
|
| 1
| ---------------- dx
| 2
| / ___ \
| |-\/ 2 |
| |-------*x| + 1 / ___\
| \ 4 / ___ |x*\/ 2 |
| \/ 2 *atan|-------|
/ \ 4 /
---------------------- = -------------------
8 4 Solution is:
/ ___\
___ |x*\/ 2 |
\/ 2 *atan|-------|
\ 4 /
C + -------------------
4
The answer (Indefinite)
[src]
/ ___\ / ___ |x*\/ 2 | | \/ 2 *atan|-------| | 1 \ 4 / | ------ dx = C + ------------------- | 2 4 | x + 8 | /
$$\int \frac{1}{x^{2} + 8}\, dx = C + \frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} x}{4} \right)}}{4}$$
The graph
The answer
[src]
/ ___\
___ |\/ 2 |
\/ 2 *atan|-----|
\ 4 /
-----------------
4
$$\frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2}}{4} \right)}}{4}$$
=
=
/ ___\
___ |\/ 2 |
\/ 2 *atan|-----|
\ 4 /
-----------------
4
$$\frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2}}{4} \right)}}{4}$$
sqrt(2)*atan(sqrt(2)/4)/4
The graph
Use the examples entering the upper and lower limits of integration.