Sr Examen
Integral de (7x-2)/(2x²+9) dx
Solución
Ha introducido
[src]
1 / | | 7*x - 2 | -------- dx | 2 | 2*x + 9 | / 0
$$\int\limits_{0}^{1} \frac{7 x - 2}{2 x^{2} + 9}\, dx$$
Integral((7*x - 2)/(2*x^2 + 9), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 7*x - 2 | -------- dx | 2 | 2*x + 9 | /
Reescribimos la función subintegral
2*2*x
7*-------------- /-2 \
2 |---|
7*x - 2 2*x + 0*x + 9 \ 9 /
-------- = ---------------- + ----------------
2 4 2
2*x + 9 / ___ \
|-\/ 2 |
|-------*x| + 1
\ 3 / o
/ | | 7*x - 2 | -------- dx | 2 = | 2*x + 9 | /
/
|
| 1
2* | ---------------- dx /
| 2 |
| / ___ \ | 2*2*x
| |-\/ 2 | 7* | -------------- dx
| |-------*x| + 1 | 2
| \ 3 / | 2*x + 0*x + 9
| |
/ /
- ------------------------ + ----------------------
9 4 En integral
/
|
| 2*2*x
7* | -------------- dx
| 2
| 2*x + 0*x + 9
|
/
----------------------
4 hacemos el cambio
2 u = 2*x
entonces
integral =
/
|
| 1
7* | ----- du
| 9 + u
|
/ 7*log(9 + u)
------------- = ------------
4 4 hacemos cambio inverso
/
|
| 2*2*x
7* | -------------- dx
| 2
| 2*x + 0*x + 9
| / 2\
/ 7*log\9 + 2*x /
---------------------- = ---------------
4 4 En integral
/
|
| 1
-2* | ---------------- dx
| 2
| / ___ \
| |-\/ 2 |
| |-------*x| + 1
| \ 3 /
|
/
-------------------------
9 hacemos el cambio
___
-x*\/ 2
v = ---------
3 entonces
integral =
/
|
| 1
-2* | ------ dv
| 2
| 1 + v
|
/ -2*atan(v)
--------------- = ----------
9 9 hacemos cambio inverso
/
|
| 1
-2* | ---------------- dx
| 2
| / ___ \
| |-\/ 2 |
| |-------*x| + 1 / ___\
| \ 3 / ___ |x*\/ 2 |
| -\/ 2 *atan|-------|
/ \ 3 /
------------------------- = ---------------------
9 3 La solución:
/ ___\
/9 2\ ___ |x*\/ 2 |
7*log|- + x | \/ 2 *atan|-------|
\2 / \ 3 /
C + ------------- - -------------------
4 3
Respuesta (Indefinida)
[src]
/ ___\ / ___ |x*\/ 2 | | / 2\ \/ 2 *atan|-------| | 7*x - 2 7*log\9 + 2*x / \ 3 / | -------- dx = C + --------------- - ------------------- | 2 4 3 | 2*x + 9 | /
$$\int \frac{7 x - 2}{2 x^{2} + 9}\, dx = C + \frac{7 \log{\left(2 x^{2} + 9 \right)}}{4} - \frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} x}{3} \right)}}{3}$$
Gráfica
Respuesta
[src]
/ ___\
___ |\/ 2 |
\/ 2 *atan|-----|
7*log(9/2) 7*log(11/2) \ 3 /
- ---------- + ----------- - -----------------
4 4 3
$$- \frac{7 \log{\left(\frac{9}{2} \right)}}{4} - \frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2}}{3} \right)}}{3} + \frac{7 \log{\left(\frac{11}{2} \right)}}{4}$$
=
=
/ ___\
___ |\/ 2 |
\/ 2 *atan|-----|
7*log(9/2) 7*log(11/2) \ 3 /
- ---------- + ----------- - -----------------
4 4 3
$$- \frac{7 \log{\left(\frac{9}{2} \right)}}{4} - \frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2}}{3} \right)}}{3} + \frac{7 \log{\left(\frac{11}{2} \right)}}{4}$$
-7*log(9/2)/4 + 7*log(11/2)/4 - sqrt(2)*atan(sqrt(2)/3)/3
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.