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- Improper integral
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How to use it?
- Integral of d{x}:
-
Integral of e^(4*x)*dx
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Integral of 1/6x
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Integral of x/(sinx^2)^2
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Integral of x*ln(x^2+1)
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Identical expressions
- one /(6x^ two + five)
- 1 divide by (6x squared plus 5)
- one divide by (6x to the power of two plus five)
- 1/(6x2+5)
- 1/6x2+5
- 1/(6x²+5)
- 1/(6x to the power of 2+5)
- 1/6x^2+5
- 1 divide by (6x^2+5)
- 1/(6x^2+5)dx
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Similar expressions
- 1/(6x^2-5)
Integral of 1/(6x^2+5) dx
The solution
You have entered
[src]
1 / | | 1 | -------- dx | 2 | 6*x + 5 | / 0
$$\int\limits_{0}^{1} \frac{1}{6 x^{2} + 5}\, dx$$
Integral(1/(6*x^2 + 5), (x, 0, 1))
Detail solution
We have the integral:
/ | | 1 | -------- dx | 2 | 6*x + 5 | /
Rewrite the integrand
1 1
-------- = ---------------------
2 / 2 \
6*x + 5 |/ ____ \ |
||-\/ 30 | |
5*||--------*x| + 1|
\\ 5 / /or
/ | | 1 | -------- dx | 2 = | 6*x + 5 | /
/
|
| 1
| ----------------- dx
| 2
| / ____ \
| |-\/ 30 |
| |--------*x| + 1
| \ 5 /
|
/
-----------------------
5 In the integral
/
|
| 1
| ----------------- dx
| 2
| / ____ \
| |-\/ 30 |
| |--------*x| + 1
| \ 5 /
|
/
-----------------------
5 do replacement
____
-x*\/ 30
v = ----------
5 then
the integral =
/
|
| 1
| ------ dv
| 2
| 1 + v
|
/ atan(v)
------------ = -------
5 5 do backward replacement
/
|
| 1
| ----------------- dx
| 2
| / ____ \
| |-\/ 30 |
| |--------*x| + 1 / ____\
| \ 5 / ____ |x*\/ 30 |
| \/ 30 *atan|--------|
/ \ 5 /
----------------------- = ---------------------
5 30 Solution is:
/ ____\
____ |x*\/ 30 |
\/ 30 *atan|--------|
\ 5 /
C + ---------------------
30
The answer (Indefinite)
[src]
/ ____\ / ____ |x*\/ 30 | | \/ 30 *atan|--------| | 1 \ 5 / | -------- dx = C + --------------------- | 2 30 | 6*x + 5 | /
$$\int \frac{1}{6 x^{2} + 5}\, dx = C + \frac{\sqrt{30} \operatorname{atan}{\left(\frac{\sqrt{30} x}{5} \right)}}{30}$$
The graph
The answer
[src]
/ ____\
____ |\/ 30 |
\/ 30 *atan|------|
\ 5 /
-------------------
30
$$\frac{\sqrt{30} \operatorname{atan}{\left(\frac{\sqrt{30}}{5} \right)}}{30}$$
=
=
/ ____\
____ |\/ 30 |
\/ 30 *atan|------|
\ 5 /
-------------------
30
$$\frac{\sqrt{30} \operatorname{atan}{\left(\frac{\sqrt{30}}{5} \right)}}{30}$$
sqrt(30)*atan(sqrt(30)/5)/30
The graph
Use the examples entering the upper and lower limits of integration.