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-
How to use it?
- Integral of d{x}:
-
Integral of x*exp(-4*x)
-
Integral of 1/(x^2-2*x)
-
Integral of (x*x)*exp(-x/2)
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Integral of x^3*ln(2x)
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Identical expressions
- (x)/((x^ two +12x+ forty-five))
- (x) divide by ((x squared plus 12x plus 45))
- (x) divide by ((x to the power of two plus 12x plus forty minus five))
- (x)/((x2+12x+45))
- x/x2+12x+45
- (x)/((x²+12x+45))
- (x)/((x to the power of 2+12x+45))
- x/x^2+12x+45
- (x) divide by ((x^2+12x+45))
- (x)/((x^2+12x+45))dx
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Similar expressions
- (x)/((x^2-12x+45))
- (x)/((x^2+12x-45))
Integral of (x)/((x^2+12x+45)) dx
The solution
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[src]
1 / | | x | -------------- dx | 2 | x + 12*x + 45 | / 0
$$\int\limits_{0}^{1} \frac{x}{\left(x^{2} + 12 x\right) + 45}\, dx$$
Integral(x/(x^2 + 12*x + 45), (x, 0, 1))
Detail solution
We have the integral:
/ | | x | -------------- dx | 2 | x + 12*x + 45 | /
Rewrite the integrand
/ 2*x + 12 \
|--------------| /-6 \
| 2 | |---|
x \x + 12*x + 45/ \ 9 /
-------------- = ---------------- + --------------
2 2 2
x + 12*x + 45 / x \
|- - - 2| + 1
\ 3 / or
/ | | x | -------------- dx | 2 = | x + 12*x + 45 | /
/
|
/ | 1
| 2* | -------------- dx
| 2*x + 12 | 2
| -------------- dx | / x \
| 2 | |- - - 2| + 1
| x + 12*x + 45 | \ 3 /
| |
/ /
-------------------- - ----------------------
2 3 In the integral
/
|
| 2*x + 12
| -------------- dx
| 2
| x + 12*x + 45
|
/
--------------------
2 do replacement
2 u = x + 12*x
then
the integral =
/
|
| 1
| ------ du
| 45 + u
|
/ log(45 + u)
------------ = -----------
2 2 do backward replacement
/
|
| 2*x + 12
| -------------- dx
| 2
| x + 12*x + 45
| / 2 \
/ log\45 + x + 12*x/
-------------------- = -------------------
2 2 In the integral
/
|
| 1
-2* | -------------- dx
| 2
| / x \
| |- - - 2| + 1
| \ 3 /
|
/
-----------------------
3 do replacement
x
v = -2 - -
3then
the integral =
/
|
| 1
-2* | ------ dv
| 2
| 1 + v
|
/ -2*atan(v)
--------------- = ----------
3 3 do backward replacement
/
|
| 1
-2* | -------------- dx
| 2
| / x \
| |- - - 2| + 1
| \ 3 /
|
/ / x\
----------------------- = -2*atan|2 + -|
3 \ 3/Solution is:
/ 2 \
log\45 + x + 12*x/ / x\
C + ------------------- - 2*atan|2 + -|
2 \ 3/
The answer (Indefinite)
[src]
/ | / 2 \ | x log\45 + x + 12*x/ / x\ | -------------- dx = C + ------------------- - 2*atan|2 + -| | 2 2 \ 3/ | x + 12*x + 45 | /
$$\int \frac{x}{\left(x^{2} + 12 x\right) + 45}\, dx = C + \frac{\log{\left(x^{2} + 12 x + 45 \right)}}{2} - 2 \operatorname{atan}{\left(\frac{x}{3} + 2 \right)}$$
The graph
The answer
[src]
log(58) log(45) ------- - 2*atan(7/3) + 2*atan(2) - ------- 2 2
$$- 2 \operatorname{atan}{\left(\frac{7}{3} \right)} - \frac{\log{\left(45 \right)}}{2} + \frac{\log{\left(58 \right)}}{2} + 2 \operatorname{atan}{\left(2 \right)}$$
=
=
log(58) log(45) ------- - 2*atan(7/3) + 2*atan(2) - ------- 2 2
$$- 2 \operatorname{atan}{\left(\frac{7}{3} \right)} - \frac{\log{\left(45 \right)}}{2} + \frac{\log{\left(58 \right)}}{2} + 2 \operatorname{atan}{\left(2 \right)}$$
log(58)/2 - 2*atan(7/3) + 2*atan(2) - log(45)/2
Use the examples entering the upper and lower limits of integration.