Sr Examen
Integral de cos(x)/(3+cos(x)) dx
Solución
Ha introducido
[src]
pi -- 2 / | | cos(x) | ---------- dx | 3 + cos(x) | / pi -- 30
$$\int\limits_{\frac{\pi}{30}}^{\frac{\pi}{2}} \frac{\cos{\left(x \right)}}{\cos{\left(x \right)} + 3}\, dx$$
Integral(cos(x)/(3 + cos(x)), (x, pi/30, pi/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]
/ / / ___________ \\\
| | | / ___ / ___\|||
| | | / 5 \/ 5 ___ |1 \/ 5 ||||
| | | / - - ----- \/ 3 *|- + -----||||
| | | \/ 8 8 \4 4 /|||
| | | ---------------- + -----------------|||
| | ___ | 1 2 2 |||
| |\/ 2 *|------------------------------------ - ------------------------------------|||
| | | ___________ ___________|||
| | | / ___ / ___ |||
| | | ___ / 5 \/ 5 ___ / 5 \/ 5 |||
| | | ___ \/ 3 * / - - ----- ___ \/ 3 * / - - ----- |||
| | | 1 \/ 5 \/ 8 8 1 \/ 5 \/ 8 8 |||
/ / ___\\ | | |- - - ----- + ---------------------- - - - ----- + ----------------------|||
___ | |\/ 2 || ___ | | \ 8 8 2 8 8 2 /||
3*\/ 2 *|-pi + atan|-----|| 3*\/ 2 *|-pi + atan|-----------------------------------------------------------------------------------||
7*pi \ \ 2 // \ \ 2 //
---- - --------------------------- + ---------------------------------------------------------------------------------------------------------
15 2 2
$$\frac{3 \sqrt{2} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{2} \left(- \frac{\frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{3} \left(\frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{2}}{- \frac{\sqrt{5}}{8} - \frac{1}{8} + \frac{\sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}} + \frac{1}{- \frac{\sqrt{5}}{8} - \frac{1}{8} + \frac{\sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}}\right)}{2} \right)}\right)}{2} + \frac{7 \pi}{15} - \frac{3 \sqrt{2} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}\right)}{2}$$
=
=
/ / / ___________ \\\
| | | / ___ / ___\|||
| | | / 5 \/ 5 ___ |1 \/ 5 ||||
| | | / - - ----- \/ 3 *|- + -----||||
| | | \/ 8 8 \4 4 /|||
| | | ---------------- + -----------------|||
| | ___ | 1 2 2 |||
| |\/ 2 *|------------------------------------ - ------------------------------------|||
| | | ___________ ___________|||
| | | / ___ / ___ |||
| | | ___ / 5 \/ 5 ___ / 5 \/ 5 |||
| | | ___ \/ 3 * / - - ----- ___ \/ 3 * / - - ----- |||
| | | 1 \/ 5 \/ 8 8 1 \/ 5 \/ 8 8 |||
/ / ___\\ | | |- - - ----- + ---------------------- - - - ----- + ----------------------|||
___ | |\/ 2 || ___ | | \ 8 8 2 8 8 2 /||
3*\/ 2 *|-pi + atan|-----|| 3*\/ 2 *|-pi + atan|-----------------------------------------------------------------------------------||
7*pi \ \ 2 // \ \ 2 //
---- - --------------------------- + ---------------------------------------------------------------------------------------------------------
15 2 2
$$\frac{3 \sqrt{2} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{2} \left(- \frac{\frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{3} \left(\frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{2}}{- \frac{\sqrt{5}}{8} - \frac{1}{8} + \frac{\sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}} + \frac{1}{- \frac{\sqrt{5}}{8} - \frac{1}{8} + \frac{\sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}}\right)}{2} \right)}\right)}{2} + \frac{7 \pi}{15} - \frac{3 \sqrt{2} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}\right)}{2}$$
7*pi/15 - 3*sqrt(2)*(-pi + atan(sqrt(2)/2))/2 + 3*sqrt(2)*(-pi + atan(sqrt(2)*(1/(-1/8 - sqrt(5)/8 + sqrt(3)*sqrt(5/8 - sqrt(5)/8)/2) - (sqrt(5/8 - sqrt(5)/8)/2 + sqrt(3)*(1/4 + sqrt(5)/4)/2)/(-1/8 - sqrt(5)/8 + sqrt(3)*sqrt(5/8 - sqrt(5)/8)/2))/2))/2
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.