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- Improper integral
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How to use it?
- Integral of d{x}:
-
Integral of (dx)/(4x^2+12x+13)
- Integral of e^(xy)
- Integral of 1/sqrt(sin(x))
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Integral of xxx
-
Identical expressions
- (dx)/(4x^ two +12x+ thirteen)
- (dx) divide by (4x squared plus 12x plus 13)
- (dx) divide by (4x to the power of two plus 12x plus thirteen)
- (dx)/(4x2+12x+13)
- dx/4x2+12x+13
- (dx)/(4x²+12x+13)
- (dx)/(4x to the power of 2+12x+13)
- dx/4x^2+12x+13
- (dx) divide by (4x^2+12x+13)
-
Similar expressions
- (dx)/(4x^2-12x+13)
- (dx)/(4x^2+12x-13)
Integral of (dx)/(4x^2+12x+13) dx
The solution
You have entered
[src]
oo / | | 1 | ---------------- dx | 2 | 4*x + 12*x + 13 | / 0
$$\int\limits_{0}^{\infty} \frac{1}{\left(4 x^{2} + 12 x\right) + 13}\, dx$$
Integral(1/(4*x^2 + 12*x + 13), (x, 0, oo))
Detail solution
We have the integral:
/ | | 1 | ---------------- dx | 2 | 4*x + 12*x + 13 | /
Rewrite the integrand
1 1 ---------------- = ------------------- 2 / 2 \ 4*x + 12*x + 13 4*\(-x - 3/2) + 1/
or
/ | | 1 | ---------------- dx | 2 = | 4*x + 12*x + 13 | /
/
|
| 1
| --------------- dx
| 2
| (-x - 3/2) + 1
|
/
---------------------
4 In the integral
/
|
| 1
| --------------- dx
| 2
| (-x - 3/2) + 1
|
/
---------------------
4 do replacement
v = -3/2 - x
then
the integral =
/
|
| 1
| ------ dv
| 2
| 1 + v
|
/ atan(v)
------------ = -------
4 4 do backward replacement
/
|
| 1
| --------------- dx
| 2
| (-x - 3/2) + 1
|
/ atan(3/2 + x)
--------------------- = -------------
4 4 Solution is:
atan(3/2 + x)
C + -------------
4
The answer (Indefinite)
[src]
/ | | 1 atan(3/2 + x) | ---------------- dx = C + ------------- | 2 4 | 4*x + 12*x + 13 | /
$$\int \frac{1}{\left(4 x^{2} + 12 x\right) + 13}\, dx = C + \frac{\operatorname{atan}{\left(x + \frac{3}{2} \right)}}{4}$$
The graph
The answer
[src]
atan(3/2) pi
- --------- + --
4 8
$$- \frac{\operatorname{atan}{\left(\frac{3}{2} \right)}}{4} + \frac{\pi}{8}$$
=
=
atan(3/2) pi
- --------- + --
4 8
$$- \frac{\operatorname{atan}{\left(\frac{3}{2} \right)}}{4} + \frac{\pi}{8}$$
-atan(3/2)/4 + pi/8
The graph
Use the examples entering the upper and lower limits of integration.