Sr Examen
Integral de cosx/(3+cosx) dx
Solución
Ha introducido
[src]
___
pi*\/ 2
--------
2
/
|
| cos(x)
| ---------- dx
| 3 + cos(x)
|
/
pi
--
3
$$\int\limits_{\frac{\pi}{3}}^{\frac{\sqrt{2} \pi}{2}} \frac{\cos{\left(x \right)}}{\cos{\left(x \right)} + 3}\, dx$$
Integral(cos(x)/(3 + cos(x)), (x, pi/3, pi*sqrt(2)/2))
Respuesta (Indefinida)
[src]
/ /x pi\ / ___ /x\\\
| |- - --| |\/ 2 *tan|-|||
/ ___ | |2 2 | | \2/||
| 3*\/ 2 *|pi*floor|------| + atan|------------||
| cos(x) \ \ pi / \ 2 //
| ---------- dx = C + x - -----------------------------------------------
| 3 + cos(x) 2
|
/
$$\int \frac{\cos{\left(x \right)}}{\cos{\left(x \right)} + 3}\, dx = C + x - \frac{3 \sqrt{2} \left(\operatorname{atan}{\left(\frac{\sqrt{2} \tan{\left(\frac{x}{2} \right)}}{2} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{2}$$
Gráfica
Respuesta
[src]
/ / / ___\\\
| | ___ |pi*\/ 2 |||
| |\/ 2 *tan|--------||| / / ___\\
___ | | \ 4 /|| ___ | |\/ 6 ||
___ 3*\/ 2 *|-pi + atan|-------------------|| 3*\/ 2 *|-pi + atan|-----||
pi pi*\/ 2 \ \ 2 // \ \ 6 //
- -- + -------- - ----------------------------------------- + ---------------------------
3 2 2 2
$$\frac{3 \sqrt{2} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{6}}{6} \right)}\right)}{2} - \frac{\pi}{3} + \frac{\sqrt{2} \pi}{2} - \frac{3 \sqrt{2} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{2} \tan{\left(\frac{\sqrt{2} \pi}{4} \right)}}{2} \right)}\right)}{2}$$
=
=
/ / / ___\\\
| | ___ |pi*\/ 2 |||
| |\/ 2 *tan|--------||| / / ___\\
___ | | \ 4 /|| ___ | |\/ 6 ||
___ 3*\/ 2 *|-pi + atan|-------------------|| 3*\/ 2 *|-pi + atan|-----||
pi pi*\/ 2 \ \ 2 // \ \ 6 //
- -- + -------- - ----------------------------------------- + ---------------------------
3 2 2 2
$$\frac{3 \sqrt{2} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{6}}{6} \right)}\right)}{2} - \frac{\pi}{3} + \frac{\sqrt{2} \pi}{2} - \frac{3 \sqrt{2} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{2} \tan{\left(\frac{\sqrt{2} \pi}{4} \right)}}{2} \right)}\right)}{2}$$
-pi/3 + pi*sqrt(2)/2 - 3*sqrt(2)*(-pi + atan(sqrt(2)*tan(pi*sqrt(2)/4)/2))/2 + 3*sqrt(2)*(-pi + atan(sqrt(6)/6))/2
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.