Sr Examen
Integral de (2-x)/(4x^2+12x+11) dx
Solución
Ha introducido
[src]
1 / | | 2 - x | ---------------- dx | 2 | 4*x + 12*x + 11 | / 0
$$\int\limits_{0}^{1} \frac{2 - x}{\left(4 x^{2} + 12 x\right) + 11}\, dx$$
Integral((2 - x)/(4*x^2 + 12*x + 11), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 2 - x | ---------------- dx | 2 | 4*x + 12*x + 11 | /
Reescribimos la función subintegral
/ 4*2*x + 12 \
|----------------| / 7 \
| 2 | |---|
2 - x \4*x + 12*x + 11/ \2*2/
---------------- = - ------------------ + -------------------------
2 8 2
4*x + 12*x + 11 / ___\
| ___ 3*\/ 2 |
|-\/ 2 *x - -------| + 1
\ 2 / o
/ | | 2 - x | ---------------- dx | 2 = | 4*x + 12*x + 11 | /
/
|
| 1
/ 7* | ------------------------- dx
| | 2
| 4*2*x + 12 | / ___\
| ---------------- dx | | ___ 3*\/ 2 |
| 2 | |-\/ 2 *x - -------| + 1
| 4*x + 12*x + 11 | \ 2 /
| |
/ /
- ---------------------- + ---------------------------------
8 4 En integral
/
|
| 4*2*x + 12
- | ---------------- dx
| 2
| 4*x + 12*x + 11
|
/
------------------------
8 hacemos el cambio
2 u = 4*x + 12*x
entonces
integral =
/
|
| 1
- | ------ du
| 11 + u
|
/ -log(11 + u)
-------------- = -------------
8 8 hacemos cambio inverso
/
|
| 4*2*x + 12
- | ---------------- dx
| 2
| 4*x + 12*x + 11
| / 2 \
/ -log\11 + 4*x + 12*x/
------------------------ = -----------------------
8 8 En integral
/
|
| 1
7* | ------------------------- dx
| 2
| / ___\
| | ___ 3*\/ 2 |
| |-\/ 2 *x - -------| + 1
| \ 2 /
|
/
---------------------------------
4 hacemos el cambio
___
3*\/ 2 ___
v = - ------- - x*\/ 2
2 entonces
integral =
/
|
| 1
7* | ------ dv
| 2
| 1 + v
|
/ 7*atan(v)
-------------- = ---------
4 4 hacemos cambio inverso
/
|
| 1
7* | ------------------------- dx
| 2
| / ___\
| | ___ 3*\/ 2 |
| |-\/ 2 *x - -------| + 1 / ___ \
| \ 2 / ___ |3*\/ 2 ___|
| 7*\/ 2 *atan|------- + x*\/ 2 |
/ \ 2 /
--------------------------------- = -------------------------------
4 8 La solución:
/ ___ \
/11 2 \ ___ |3*\/ 2 ___|
log|-- + x + 3*x| 7*\/ 2 *atan|------- + x*\/ 2 |
\4 / \ 2 /
C - ------------------ + -------------------------------
8 8
Respuesta (Indefinida)
[src]
/ | / 2 \ ___ / ___ \ | 2 - x log\11 + 4*x + 12*x/ 7*\/ 2 *atan\\/ 2 *(3/2 + x)/ | ---------------- dx = C - --------------------- + ----------------------------- | 2 8 8 | 4*x + 12*x + 11 | /
$$\int \frac{2 - x}{\left(4 x^{2} + 12 x\right) + 11}\, dx = C - \frac{\log{\left(4 x^{2} + 12 x + 11 \right)}}{8} + \frac{7 \sqrt{2} \operatorname{atan}{\left(\sqrt{2} \left(x + \frac{3}{2}\right) \right)}}{8}$$
Gráfica
Respuesta
[src]
/ ___\ / ___\
___ |3*\/ 2 | ___ |5*\/ 2 |
7*\/ 2 *atan|-------| 7*\/ 2 *atan|-------|
log(27/4) log(11/4) \ 2 / \ 2 /
- --------- + --------- - --------------------- + ---------------------
8 8 8 8
$$- \frac{7 \sqrt{2} \operatorname{atan}{\left(\frac{3 \sqrt{2}}{2} \right)}}{8} - \frac{\log{\left(\frac{27}{4} \right)}}{8} + \frac{\log{\left(\frac{11}{4} \right)}}{8} + \frac{7 \sqrt{2} \operatorname{atan}{\left(\frac{5 \sqrt{2}}{2} \right)}}{8}$$
=
=
/ ___\ / ___\
___ |3*\/ 2 | ___ |5*\/ 2 |
7*\/ 2 *atan|-------| 7*\/ 2 *atan|-------|
log(27/4) log(11/4) \ 2 / \ 2 /
- --------- + --------- - --------------------- + ---------------------
8 8 8 8
$$- \frac{7 \sqrt{2} \operatorname{atan}{\left(\frac{3 \sqrt{2}}{2} \right)}}{8} - \frac{\log{\left(\frac{27}{4} \right)}}{8} + \frac{\log{\left(\frac{11}{4} \right)}}{8} + \frac{7 \sqrt{2} \operatorname{atan}{\left(\frac{5 \sqrt{2}}{2} \right)}}{8}$$
-log(27/4)/8 + log(11/4)/8 - 7*sqrt(2)*atan(3*sqrt(2)/2)/8 + 7*sqrt(2)*atan(5*sqrt(2)/2)/8
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.