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How to use it?
- Integral of d{x}:
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Integral of e^(4*x)*dx
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Integral of 1/(x^2-9)
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Integral of 2*x/(-1+x)
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Integral of 1/(2+3x^2)
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Identical expressions
- one /(two +3x^ two)
- 1 divide by (2 plus 3x squared )
- one divide by (two plus 3x to the power of two)
- 1/(2+3x2)
- 1/2+3x2
- 1/(2+3x²)
- 1/(2+3x to the power of 2)
- 1/2+3x^2
- 1 divide by (2+3x^2)
- 1/(2+3x^2)dx
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Similar expressions
- 1/(2-3x^2)
Integral of 1/(2+3x^2) dx
The solution
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[src]
1 / | | 1 | -------- dx | 2 | 2 + 3*x | / 0
$$\int\limits_{0}^{1} \frac{1}{3 x^{2} + 2}\, dx$$
Integral(1/(2 + 3*x^2), (x, 0, 1))
Detail solution
We have the integral:
/ | | 1 | -------- dx | 2 | 2 + 3*x | /
Rewrite the integrand
1 1
-------- = --------------------
2 / 2 \
2 + 3*x |/ ___ \ |
||-\/ 6 | |
2*||-------*x| + 1|
\\ 2 / /or
/ | | 1 | -------- dx | 2 = | 2 + 3*x | /
/
|
| 1
| ---------------- dx
| 2
| / ___ \
| |-\/ 6 |
| |-------*x| + 1
| \ 2 /
|
/
----------------------
2 In the integral
/
|
| 1
| ---------------- dx
| 2
| / ___ \
| |-\/ 6 |
| |-------*x| + 1
| \ 2 /
|
/
----------------------
2 do replacement
___
-x*\/ 6
v = ---------
2 then
the integral =
/
|
| 1
| ------ dv
| 2
| 1 + v
|
/ atan(v)
------------ = -------
2 2 do backward replacement
/
|
| 1
| ---------------- dx
| 2
| / ___ \
| |-\/ 6 |
| |-------*x| + 1 / ___\
| \ 2 / ___ |x*\/ 6 |
| \/ 6 *atan|-------|
/ \ 2 /
---------------------- = -------------------
2 6 Solution is:
/ ___\
___ |x*\/ 6 |
\/ 6 *atan|-------|
\ 2 /
C + -------------------
6
The answer (Indefinite)
[src]
/ ___\ / ___ |x*\/ 6 | | \/ 6 *atan|-------| | 1 \ 2 / | -------- dx = C + ------------------- | 2 6 | 2 + 3*x | /
$$\int \frac{1}{3 x^{2} + 2}\, dx = C + \frac{\sqrt{6} \operatorname{atan}{\left(\frac{\sqrt{6} x}{2} \right)}}{6}$$
The graph
The answer
[src]
/ ___\
___ |\/ 6 |
\/ 6 *atan|-----|
\ 2 /
-----------------
6
$$\frac{\sqrt{6} \operatorname{atan}{\left(\frac{\sqrt{6}}{2} \right)}}{6}$$
=
=
/ ___\
___ |\/ 6 |
\/ 6 *atan|-----|
\ 2 /
-----------------
6
$$\frac{\sqrt{6} \operatorname{atan}{\left(\frac{\sqrt{6}}{2} \right)}}{6}$$
sqrt(6)*atan(sqrt(6)/2)/6
The graph
Use the examples entering the upper and lower limits of integration.