Sr Examen
Integral de (4*1)/(x^2+5*x+10) dx
Solución
Ha introducido
[src]
1 / | | 4 | ------------- dx | 2 | x + 5*x + 10 | / 0
$$\int\limits_{0}^{1} \frac{4}{\left(x^{2} + 5 x\right) + 10}\, dx$$
Integral(4/(x^2 + 5*x + 10), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 4 | ------------- dx | 2 | x + 5*x + 10 | /
Reescribimos la función subintegral
/ 4 \
|----|
4 \15/4/
------------- = ---------------------------
2 2
x + 5*x + 10 / ____ ____\
|-2*\/ 15 \/ 15 |
|---------*x - ------| + 1
\ 15 3 / o
/ | | 4 | ------------- dx | 2 = | x + 5*x + 10 | /
/
|
| 1
16* | --------------------------- dx
| 2
| / ____ ____\
| |-2*\/ 15 \/ 15 |
| |---------*x - ------| + 1
| \ 15 3 /
|
/
------------------------------------
15 En integral
/
|
| 1
16* | --------------------------- dx
| 2
| / ____ ____\
| |-2*\/ 15 \/ 15 |
| |---------*x - ------| + 1
| \ 15 3 /
|
/
------------------------------------
15 hacemos el cambio
____ ____
\/ 15 2*x*\/ 15
v = - ------ - ----------
3 15 entonces
integral =
/
|
| 1
16* | ------ dv
| 2
| 1 + v
|
/ 16*atan(v)
--------------- = ----------
15 15 hacemos cambio inverso
/
|
| 1
16* | --------------------------- dx
| 2
| / ____ ____\
| |-2*\/ 15 \/ 15 |
| |---------*x - ------| + 1 / ____ ____\
| \ 15 3 / ____ |\/ 15 2*x*\/ 15 |
| 8*\/ 15 *atan|------ + ----------|
/ \ 3 15 /
------------------------------------ = ----------------------------------
15 15 La solución:
/ ____ ____\
____ |\/ 15 2*x*\/ 15 |
8*\/ 15 *atan|------ + ----------|
\ 3 15 /
C + ----------------------------------
15
Respuesta (Indefinida)
[src]
/ ____ \ / ____ |2*\/ 15 *(5/2 + x)| | 8*\/ 15 *atan|------------------| | 4 \ 15 / | ------------- dx = C + --------------------------------- | 2 15 | x + 5*x + 10 | /
$$\int \frac{4}{\left(x^{2} + 5 x\right) + 10}\, dx = C + \frac{8 \sqrt{15} \operatorname{atan}{\left(\frac{2 \sqrt{15} \left(x + \frac{5}{2}\right)}{15} \right)}}{15}$$
Gráfica
Respuesta
[src]
/ ____\ / ____\
____ |\/ 15 | ____ |7*\/ 15 |
8*\/ 15 *atan|------| 8*\/ 15 *atan|--------|
\ 3 / \ 15 /
- --------------------- + -----------------------
15 15
$$- \frac{8 \sqrt{15} \operatorname{atan}{\left(\frac{\sqrt{15}}{3} \right)}}{15} + \frac{8 \sqrt{15} \operatorname{atan}{\left(\frac{7 \sqrt{15}}{15} \right)}}{15}$$
=
=
/ ____\ / ____\
____ |\/ 15 | ____ |7*\/ 15 |
8*\/ 15 *atan|------| 8*\/ 15 *atan|--------|
\ 3 / \ 15 /
- --------------------- + -----------------------
15 15
$$- \frac{8 \sqrt{15} \operatorname{atan}{\left(\frac{\sqrt{15}}{3} \right)}}{15} + \frac{8 \sqrt{15} \operatorname{atan}{\left(\frac{7 \sqrt{15}}{15} \right)}}{15}$$
-8*sqrt(15)*atan(sqrt(15)/3)/15 + 8*sqrt(15)*atan(7*sqrt(15)/15)/15
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.