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How to use it?
- Integral of d{x}:
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Integral of e^(2*x)/x
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Integral of ×
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Integral of 1/(x^1/3)
- Integral of (1+x)/x^2
- How do you in partial fractions?:
- 1/(x^2+4x+7)
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Identical expressions
- one /(x^ two +4x+ seven)
- 1 divide by (x squared plus 4x plus 7)
- one divide by (x to the power of two plus 4x plus seven)
- 1/(x2+4x+7)
- 1/x2+4x+7
- 1/(x²+4x+7)
- 1/(x to the power of 2+4x+7)
- 1/x^2+4x+7
- 1 divide by (x^2+4x+7)
- 1/(x^2+4x+7)dx
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Similar expressions
- 1/(x^2+4x-7)
- 1/(x^2-4x+7)
Integral of 1/(x^2+4x+7) dx
The solution
You have entered
[src]
1 / | | 1 | ------------ dx | 2 | x + 4*x + 7 | / -3
$$\int\limits_{-3}^{1} \frac{1}{\left(x^{2} + 4 x\right) + 7}\, dx$$
Integral(1/(x^2 + 4*x + 7), (x, -3, 1))
Detail solution
We have the integral:
/ | | 1 | ------------ dx | 2 | x + 4*x + 7 | /
Rewrite the integrand
1 1
------------ = ------------------------------
2 / 2 \
x + 4*x + 7 |/ ___ ___\ |
||-\/ 3 2*\/ 3 | |
3*||-------*x - -------| + 1|
\\ 3 3 / /or
/ | | 1 | ------------ dx | 2 = | x + 4*x + 7 | /
/
|
| 1
| -------------------------- dx
| 2
| / ___ ___\
| |-\/ 3 2*\/ 3 |
| |-------*x - -------| + 1
| \ 3 3 /
|
/
--------------------------------
3 In the integral
/
|
| 1
| -------------------------- dx
| 2
| / ___ ___\
| |-\/ 3 2*\/ 3 |
| |-------*x - -------| + 1
| \ 3 3 /
|
/
--------------------------------
3 do replacement
___ ___
2*\/ 3 x*\/ 3
v = - ------- - -------
3 3 then
the integral =
/
|
| 1
| ------ dv
| 2
| 1 + v
|
/ atan(v)
------------ = -------
3 3 do backward replacement
/
|
| 1
| -------------------------- dx
| 2
| / ___ ___\
| |-\/ 3 2*\/ 3 |
| |-------*x - -------| + 1 / ___ ___\
| \ 3 3 / ___ |2*\/ 3 x*\/ 3 |
| \/ 3 *atan|------- + -------|
/ \ 3 3 /
-------------------------------- = -----------------------------
3 3 Solution is:
/ ___ ___\
___ |2*\/ 3 x*\/ 3 |
\/ 3 *atan|------- + -------|
\ 3 3 /
C + -----------------------------
3
The answer (Indefinite)
[src]
/ ___ \ / ___ |\/ 3 *(2 + x)| | \/ 3 *atan|-------------| | 1 \ 3 / | ------------ dx = C + ------------------------- | 2 3 | x + 4*x + 7 | /
$$\int \frac{1}{\left(x^{2} + 4 x\right) + 7}\, dx = C + \frac{\sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \left(x + 2\right)}{3} \right)}}{3}$$
The graph
The answer
[src]
___ pi*\/ 3 -------- 6
$$\frac{\sqrt{3} \pi}{6}$$
=
=
___ pi*\/ 3 -------- 6
$$\frac{\sqrt{3} \pi}{6}$$
pi*sqrt(3)/6
Use the examples entering the upper and lower limits of integration.