Sr Examen
Integral de t/(t^2+2*t+4)dt dt
Solución
Ha introducido
[src]
1 / | | t | ------------ dt | 2 | t + 2*t + 4 | / 0
$$\int\limits_{0}^{1} \frac{t}{\left(t^{2} + 2 t\right) + 4}\, dt$$
Integral(t/(t^2 + 2*t + 4), (t, 0, 1))
Solución detallada
Tenemos el integral:
/ | | t | ------------ dt | 2 | t + 2*t + 4 | /
Reescribimos la función subintegral
/ 2*t + 2 \
|------------| /-1 \
| 2 | |---|
t \t + 2*t + 4/ \ 3 /
------------ = -------------- + ------------------------
2 2 2
t + 2*t + 4 / ___ ___\
|-\/ 3 \/ 3 |
|-------*t - -----| + 1
\ 3 3 / o
/ | | t | ------------ dt | 2 = | t + 2*t + 4 | /
/
|
| 1
/ | ------------------------ dt
| | 2
| 2*t + 2 | / ___ ___\
| ------------ dt | |-\/ 3 \/ 3 |
| 2 | |-------*t - -----| + 1
| t + 2*t + 4 | \ 3 3 /
| |
/ /
------------------ - ------------------------------
2 3 En integral
/
|
| 2*t + 2
| ------------ dt
| 2
| t + 2*t + 4
|
/
------------------
2 hacemos el cambio
2 u = t + 2*t
entonces
integral =
/
|
| 1
| ----- du
| 4 + u
|
/ log(4 + u)
----------- = ----------
2 2 hacemos cambio inverso
/
|
| 2*t + 2
| ------------ dt
| 2
| t + 2*t + 4
| / 2 \
/ log\4 + t + 2*t/
------------------ = -----------------
2 2 En integral
/
|
| 1
- | ------------------------ dt
| 2
| / ___ ___\
| |-\/ 3 \/ 3 |
| |-------*t - -----| + 1
| \ 3 3 /
|
/
--------------------------------
3 hacemos el cambio
___ ___
\/ 3 t*\/ 3
v = - ----- - -------
3 3 entonces
integral =
/
|
| 1
- | ------ dv
| 2
| 1 + v
|
/ -atan(v)
-------------- = ---------
3 3 hacemos cambio inverso
/
|
| 1
- | ------------------------ dt
| 2
| / ___ ___\
| |-\/ 3 \/ 3 |
| |-------*t - -----| + 1 / ___ ___\
| \ 3 3 / ___ |\/ 3 t*\/ 3 |
| -\/ 3 *atan|----- + -------|
/ \ 3 3 /
-------------------------------- = -----------------------------
3 3 La solución:
/ ___ ___\
___ |\/ 3 t*\/ 3 |
/ 2 \ \/ 3 *atan|----- + -------|
log\4 + t + 2*t/ \ 3 3 /
C + ----------------- - ---------------------------
2 3
Respuesta (Indefinida)
[src]
/ ___ \ / ___ |\/ 3 *(1 + t)| | / 2 \ \/ 3 *atan|-------------| | t log\4 + t + 2*t/ \ 3 / | ------------ dt = C + ----------------- - ------------------------- | 2 2 3 | t + 2*t + 4 | /
$$\int \frac{t}{\left(t^{2} + 2 t\right) + 4}\, dt = C + \frac{\log{\left(t^{2} + 2 t + 4 \right)}}{2} - \frac{\sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \left(t + 1\right)}{3} \right)}}{3}$$
Gráfica
Respuesta
[src]
/ ___\
___ |2*\/ 3 |
\/ 3 *atan|-------| ___
log(7) log(4) \ 3 / pi*\/ 3
------ - ------ - ------------------- + --------
2 2 3 18
$$- \frac{\log{\left(4 \right)}}{2} - \frac{\sqrt{3} \operatorname{atan}{\left(\frac{2 \sqrt{3}}{3} \right)}}{3} + \frac{\sqrt{3} \pi}{18} + \frac{\log{\left(7 \right)}}{2}$$
=
=
/ ___\
___ |2*\/ 3 |
\/ 3 *atan|-------| ___
log(7) log(4) \ 3 / pi*\/ 3
------ - ------ - ------------------- + --------
2 2 3 18
$$- \frac{\log{\left(4 \right)}}{2} - \frac{\sqrt{3} \operatorname{atan}{\left(\frac{2 \sqrt{3}}{3} \right)}}{3} + \frac{\sqrt{3} \pi}{18} + \frac{\log{\left(7 \right)}}{2}$$
log(7)/2 - log(4)/2 - sqrt(3)*atan(2*sqrt(3)/3)/3 + pi*sqrt(3)/18
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.